💡Okay, so I have this equation here: x squared equals 2x plus 2. I need to find the value of (alpha minus beta) squared, where alpha and beta are the two distinct solutions to this equation. Hmm, let me think about how to approach this.First, I remember that quadratic equations have two solutions, and those solutions can be found using the quadratic formula. The standard form of a quadratic equation is ax squared plus bx plus c equals zero. So, I should probably rewrite the given equation in that form to make it easier to apply the quadratic formula.Starting with x squared equals 2x plus 2, I can subtract 2x and 2 from both sides to get everything on one side. That would give me x squared minus 2x minus 2 equals zero. Okay, so now it's in the standard quadratic form, where a is 1, b is -2, and c is -2.Now, I recall the quadratic formula: x equals negative b plus or minus the square root of (b squared minus 4ac) all over 2a. Plugging in the values I have, that would be x equals negative (-2) plus or minus the square root of ((-2) squared minus 4 times 1 times (-2)) all over 2 times 1. Let me calculate that step by step.First, negative (-2) is just 2. Then, inside the square root, (-2) squared is 4. Then, 4 times 1 times (-2) is -8. But since it's minus that, it becomes plus 8. So, the square root part is the square root of 4 plus 8, which is the square root of 12. I can simplify the square root of 12 as 2 times the square root of 3, right? Because 12 is 4 times 3, and the square root of 4 is 2.So, putting it all together, x equals (2 plus or minus 2 times the square root of 3) divided by 2. I can factor out a 2 from the numerator, which would give me 2 times (1 plus or minus the square root of 3) over 2. The 2s cancel out, so x equals 1 plus or minus the square root of 3.Okay, so the two solutions are alpha equals 1 plus the square root of 3 and beta equals 1 minus the square root of 3. Now, I need to find (alpha minus beta) squared. Let me compute alpha minus beta first.Alpha minus beta is (1 plus the square root of 3) minus (1 minus the square root of 3). Let's distribute the negative sign to the second term: 1 plus the square root of 3 minus 1 plus the square root of 3. The 1 and minus 1 cancel each other out, leaving me with the square root of 3 plus the square root of 3, which is 2 times the square root of 3.So, alpha minus beta is 2 times the square root of 3. Now, I need to square this result. Squaring 2 times the square root of 3 would be (2)^2 times (square root of 3)^2. That's 4 times 3, which equals 12.Wait, let me double-check that to make sure I didn't make a mistake. Starting from the beginning, the equation was x squared equals 2x plus 2. I rearranged it to x squared minus 2x minus 2 equals zero, which seems correct. Then, applying the quadratic formula with a=1, b=-2, c=-2, I got x equals [2 plus or minus sqrt(4 + 8)]/2, which simplifies to 1 plus or minus sqrt(3). That looks right.Then, subtracting beta from alpha: (1 + sqrt(3)) - (1 - sqrt(3)) equals 2 sqrt(3). Squaring that gives (2 sqrt(3))^2, which is 4 times 3, so 12. Yeah, that seems consistent.I guess another way to think about this without finding the actual roots is by using the properties of quadratic equations. I remember that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is -b/a and the product is c/a. So, in this case, the sum of alpha and beta is 2/1, which is 2, and the product is -2/1, which is -2.But how does that help me find (alpha - beta)^2? Hmm, maybe I can express (alpha - beta)^2 in terms of the sum and product of the roots. Let me recall the identity: (alpha - beta)^2 equals (alpha + beta)^2 minus 4 alpha beta. Is that correct? Let me check.Expanding (alpha - beta)^2, we get alpha squared minus 2 alpha beta plus beta squared. On the other hand, (alpha + beta)^2 is alpha squared plus 2 alpha beta plus beta squared. So, if I subtract 4 alpha beta from (alpha + beta)^2, I get alpha squared plus 2 alpha beta plus beta squared minus 4 alpha beta, which simplifies to alpha squared minus 2 alpha beta plus beta squared, which is exactly (alpha - beta)^2. Yes, that identity works.So, using that, I can compute (alpha - beta)^2 as (alpha + beta)^2 minus 4 alpha beta. We already know alpha + beta is 2, so (alpha + beta)^2 is 4. And alpha beta is -2, so 4 alpha beta is 4 times (-2), which is -8. Therefore, (alpha - beta)^2 is 4 minus (-8), which is 4 + 8, so 12. That matches the result I got earlier.That's reassuring. So, whether I compute it directly by finding the roots and subtracting them or use the identity involving the sum and product of the roots, I end up with the same answer, 12. I think that solidifies my confidence in the solution.Just to make sure I didn't make any arithmetic mistakes, let me go through the steps again. Starting with the equation x^2 = 2x + 2, moving everything to one side gives x^2 - 2x - 2 = 0. Quadratic formula: x = [2 ± sqrt(4 + 8)]/2 = [2 ± sqrt(12)]/2 = [2 ± 2 sqrt(3)]/2 = 1 ± sqrt(3). So, alpha is 1 + sqrt(3), beta is 1 - sqrt(3). Subtracting beta from alpha: (1 + sqrt(3)) - (1 - sqrt(3)) = 2 sqrt(3). Squaring that: (2 sqrt(3))^2 = 4 * 3 = 12.Yep, that all checks out. I think I'm good. So, the value of (alpha - beta)^2 is 12.**Final Answer**The value of (alpha - beta)^2 is boxed{12}.

💡Alright, so I've got this trigonometry problem here, and I need to figure out the value of cos(π + α) given that sin(π - α) equals log base 8 of 1/4, and α is between -π/2 and 0. Hmm, okay, let's break this down step by step.First, I remember that sin(π - α) is one of those trigonometric identities. I think it simplifies to sin α. Let me confirm that. Yeah, sin(π - α) is equal to sin α because sine is positive in the second quadrant and the identity holds. So, that means sin α equals log base 8 of 1/4.Now, I need to calculate log base 8 of 1/4. I'm not too confident about logarithms, but I think I can convert this to a more familiar base, maybe base 2, since 8 is 2 cubed and 4 is 2 squared. Let me try that.Using the change of base formula, log base 8 of 1/4 is equal to log base 2 of 1/4 divided by log base 2 of 8. Log base 2 of 1/4 is log base 2 of 2^(-2), which is -2. Log base 2 of 8 is log base 2 of 2^3, which is 3. So, log base 8 of 1/4 is -2/3. Therefore, sin α is -2/3.Okay, so sin α is -2/3. Now, I need to find cos(π + α). I remember another trigonometric identity: cos(π + α) is equal to -cos α. So, if I can find cos α, I can just take the negative of that to get the answer.To find cos α, I can use the Pythagorean identity, which says that sin² α + cos² α = 1. I know sin α is -2/3, so sin² α is (4/9). Plugging that into the identity, I get 4/9 + cos² α = 1. Subtracting 4/9 from both sides, cos² α = 5/9. Taking the square root of both sides, cos α is either sqrt(5)/3 or -sqrt(5)/3.But wait, I need to figure out the sign of cos α. The problem states that α is between -π/2 and 0, which is the fourth quadrant. In the fourth quadrant, cosine is positive, right? So, cos α must be positive. Therefore, cos α is sqrt(5)/3.Now, going back to cos(π + α), which is -cos α. Since cos α is sqrt(5)/3, then cos(π + α) is -sqrt(5)/3.Let me just double-check my steps to make sure I didn't make any mistakes. Starting from sin(π - α) = log8(1/4), I converted the logarithm to base 2 and found it equals -2/3. Then, using the identity sin(π - α) = sin α, I concluded sin α = -2/3. Using the Pythagorean identity, I found cos α = sqrt(5)/3, considering the quadrant where α lies. Finally, using the identity cos(π + α) = -cos α, I arrived at -sqrt(5)/3. Everything seems to check out.So, the value of cos(π + α) is -sqrt(5)/3, which corresponds to option B.**Final Answer**boxed{B}

💡Okay, so I have this equation here: ( x^2 - 6x + 8 = 0 ). I need to figure out if it has real roots and, if so, what they are. The options given are A) 2 and 4, B) 1 and 3, C) Non-existent, and D) 4 and 6. Hmm, let me think about how to approach this.First, I remember that for a quadratic equation of the form ( ax^2 + bx + c = 0 ), the roots can be found using the quadratic formula: ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ). The part under the square root, ( b^2 - 4ac ), is called the discriminant. The discriminant tells us about the nature of the roots. If it's positive, there are two distinct real roots; if it's zero, there's exactly one real root; and if it's negative, the roots are complex and not real.So, let's identify the coefficients in my equation. Comparing ( x^2 - 6x + 8 = 0 ) with the standard form ( ax^2 + bx + c = 0 ), I see that ( a = 1 ), ( b = -6 ), and ( c = 8 ).Now, let's calculate the discriminant: ( D = b^2 - 4ac ). Plugging in the values, that's ( (-6)^2 - 4*1*8 ). Calculating that, ( (-6)^2 = 36 ) and ( 4*1*8 = 32 ). So, ( D = 36 - 32 = 4 ). Since the discriminant is 4, which is positive, that means there are two distinct real roots. Great, so the roots are real, which rules out option C. Now, I need to find what those roots are.Using the quadratic formula: ( x = frac{-b pm sqrt{D}}{2a} ). Substituting the known values, ( x = frac{-(-6) pm sqrt{4}}{2*1} ). Simplifying that, ( x = frac{6 pm 2}{2} ). So, there are two solutions: one where we add 2 to 6 and divide by 2, and another where we subtract 2 from 6 and divide by 2. Let's calculate both:1. ( x = frac{6 + 2}{2} = frac{8}{2} = 4 )2. ( x = frac{6 - 2}{2} = frac{4}{2} = 2 )So, the roots are 4 and 2. Looking back at the options, that's option A: 2 and 4. Just to double-check, maybe I can factor the quadratic equation to see if it also gives the same roots. Factoring ( x^2 - 6x + 8 ), I need two numbers that multiply to 8 and add up to -6. Let's see: 2 and 4 multiply to 8, and 2 + 4 = 6. But since the middle term is -6x, both numbers should be negative: -2 and -4. So, the factored form is ( (x - 2)(x - 4) = 0 ), which gives the roots x = 2 and x = 4. Yep, that matches what I found earlier.I think I'm confident now that the real roots are 2 and 4, so the correct answer is A.

💡Okay, so I'm trying to figure out how to divide the 28 grams of cheese that Gray and Fat removed from their slices so that all four mice—White, Gray, Fat, and Thin—end up with equal amounts of cheese. Let me start by understanding the problem step by step.First, there are four mice dividing a cheese wheel into four slices. The slices look identical, but some have more holes, making them lighter. Specifically, Thin's slice is 20 grams lighter than Fat's slice, and White's slice is 8 grams lighter than Gray's slice. However, White isn't upset because his slice is exactly one-quarter of the total cheese weight. That means White's slice is equal to the average slice weight.Gray cut off 8 grams from his piece, and Fat cut off 20 grams from his piece. So, together, they removed 8 + 20 = 28 grams of cheese. The question is, how should these 28 grams be divided among the mice so that everyone ends up with the same amount of cheese.Let me denote the total weight of the cheese wheel as M grams. Since White's slice is one-quarter of the total weight, White's slice is M/4 grams. Gray's slice is 8 grams more than White's, so Gray's slice is M/4 + 8 grams. Similarly, Fat's slice is 20 grams more than Thin's slice. Let's denote Thin's slice as T grams, so Fat's slice is T + 20 grams.Now, the total weight of the cheese wheel should be the sum of all four slices:White's slice + Gray's slice + Fat's slice + Thin's slice = MSubstituting the expressions we have:M/4 + (M/4 + 8) + (T + 20) + T = MSimplify the equation:M/4 + M/4 + 8 + T + 20 + T = MCombine like terms:(2M)/4 + 2T + 28 = MSimplify further:M/2 + 2T + 28 = MSubtract M/2 from both sides:2T + 28 = M/2Divide both sides by 2:T + 14 = M/4So, T = M/4 - 14That means Thin's slice is M/4 - 14 grams, and Fat's slice is T + 20 = (M/4 - 14) + 20 = M/4 + 6 grams.Now, after Gray and Fat cut off their respective pieces, Gray's slice becomes (M/4 + 8) - 8 = M/4 grams, and Fat's slice becomes (M/4 + 6) - 20 = M/4 - 14 grams.So, now we have:- White: M/4 grams- Gray: M/4 grams- Fat: M/4 - 14 grams- Thin: M/4 - 14 gramsWait, that doesn't seem right. If Fat's slice after cutting is M/4 - 14 grams, and Thin's slice is also M/4 - 14 grams, then both Fat and Thin have the same amount, which is less than White and Gray. But we need all four mice to have equal amounts.I think I made a mistake in calculating Fat's slice after cutting. Let me re-examine that.Fat's original slice was T + 20 grams, which we found to be M/4 + 6 grams. After cutting off 20 grams, Fat's slice becomes (M/4 + 6) - 20 = M/4 - 14 grams. Similarly, Gray's original slice was M/4 + 8 grams, and after cutting off 8 grams, it becomes M/4 grams.So, now we have:- White: M/4 grams- Gray: M/4 grams- Fat: M/4 - 14 grams- Thin: M/4 - 14 gramsThis means that Fat and Thin each have 14 grams less than White and Gray. To equalize, we need to redistribute the 28 grams that were cut off (8 from Gray and 20 from Fat) to Fat and Thin so that everyone ends up with M/4 grams.Since Fat and Thin are each 14 grams short, we need to give each of them 14 grams. That way, Fat's slice becomes (M/4 - 14) + 14 = M/4 grams, and Thin's slice becomes (M/4 - 14) + 14 = M/4 grams. This ensures that all four mice have equal amounts of cheese.So, the 28 grams should be divided equally between Fat and Thin, with each receiving 14 grams.

💡Okay, so I have this problem where three of the four endpoints of the axes of an ellipse are given: (-3, 5), (4, -3), and (9, 5). I need to find the distance between the foci of the ellipse. Hmm, let me think about how to approach this.First, I remember that an ellipse has two axes: the major axis and the minor axis. Each axis has two endpoints, so in total, there are four endpoints. Since three of these are given, I need to figure out which one is missing and then determine the lengths of the major and minor axes to find the distance between the foci.Looking at the given points: (-3, 5), (4, -3), and (9, 5). I notice that two of these points, (-3, 5) and (9, 5), have the same y-coordinate, which is 5. That suggests they might be endpoints of the same axis, probably the major or minor axis. The third point, (4, -3), has a different y-coordinate, so it's likely part of the other axis.Let me plot these points in my mind. (-3, 5) and (9, 5) are on the same horizontal line, so they could be endpoints of the major axis. The point (4, -3) is somewhere else, so it might be an endpoint of the minor axis. But I need to confirm this.If (-3, 5) and (9, 5) are endpoints of the major axis, then the center of the ellipse would be the midpoint between these two points. Let me calculate that midpoint. The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). So, plugging in the values:Midpoint = ((-3 + 9)/2, (5 + 5)/2) = (6/2, 10/2) = (3, 5).Okay, so the center of the ellipse is at (3, 5). That makes sense because the major axis is horizontal, and the center is halfway between (-3, 5) and (9, 5).Now, the distance between (-3, 5) and (9, 5) is the length of the major axis. Let me calculate that distance. Since they have the same y-coordinate, the distance is just the difference in the x-coordinates:Distance = |9 - (-3)| = |9 + 3| = 12 units.So, the major axis is 12 units long, which means the semi-major axis (a) is half of that, so a = 6 units.Now, the other point given is (4, -3). Since the center is at (3, 5), I can find the distance from the center to this point, which should be the length of the semi-minor axis (b). Let me calculate that distance using the distance formula:Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]= sqrt[(4 - 3)^2 + (-3 - 5)^2]= sqrt[(1)^2 + (-8)^2]= sqrt[1 + 64]= sqrt[65].Wait, that's approximately 8.06 units. Hmm, but I thought the semi-minor axis should be shorter than the semi-major axis. Since a = 6, and b is sqrt(65) ≈ 8.06, which is longer than a. That doesn't make sense because in an ellipse, the semi-major axis is longer than the semi-minor axis.So, maybe I made a wrong assumption. Perhaps (-3, 5) and (9, 5) are not the endpoints of the major axis but the minor axis instead. Let me reconsider.If (-3, 5) and (9, 5) are endpoints of the minor axis, then the length of the minor axis is 12 units, so the semi-minor axis (b) is 6 units. Then, the major axis would be along the vertical direction because the point (4, -3) is not aligned horizontally with the center.Let me check the distance from the center (3, 5) to the point (4, -3) again:Distance = sqrt[(4 - 3)^2 + (-3 - 5)^2]= sqrt[1 + 64]= sqrt[65] ≈ 8.06 units.So, if this is the semi-major axis, then a = sqrt(65) ≈ 8.06, and b = 6. That makes sense because a > b in an ellipse.Now, to find the distance between the foci, I need to calculate 2c, where c is the distance from the center to each focus. I remember the relationship in an ellipse is c^2 = a^2 - b^2.Let me compute c:c^2 = a^2 - b^2= (sqrt(65))^2 - 6^2= 65 - 36= 29.So, c = sqrt(29). Therefore, the distance between the foci is 2c = 2*sqrt(29).Wait, but I thought earlier that the major axis was along the vertical direction because the point (4, -3) is not aligned horizontally. But let me double-check if (-3, 5) and (9, 5) are indeed endpoints of the minor axis.If the major axis is vertical, then the endpoints of the major axis would be (3, 5 + c) and (3, 5 - c). But I only have one point given on the vertical axis, which is (4, -3). That doesn't seem to align with the center (3, 5). Wait, maybe I made a mistake in identifying the axes.Alternatively, perhaps the major axis is horizontal, and the point (4, -3) is an endpoint of the major axis. But that would mean the major axis is not aligned horizontally, which contradicts the initial thought.Wait, let me think again. If (-3, 5) and (9, 5) are endpoints of the major axis, then the major axis is horizontal with length 12, semi-major axis a = 6. Then, the minor axis would be vertical, and the other two endpoints would be symmetric about the center (3, 5). One of them is (4, -3). So, the other endpoint should be (2, 13) because the center is (3,5), so from (4, -3), moving the same distance from the center in the opposite direction would be (3 - (4 - 3), 5 - (-3 -5)) = (2, 13). But (2,13) isn't given, so maybe that's the missing endpoint.But the problem states that three of the four endpoints are given, so the missing one is (2,13). So, the minor axis endpoints are (4, -3) and (2,13). Let me calculate the distance between these two points to confirm the length of the minor axis.Distance = sqrt[(4 - 2)^2 + (-3 -13)^2]= sqrt[(2)^2 + (-16)^2]= sqrt[4 + 256]= sqrt[260]= 2*sqrt(65).Wait, that's approximately 16.12 units, which would make the semi-minor axis b = sqrt(65) ≈ 8.06 units. But earlier, I thought a = 6, which would make b longer than a, which isn't possible because in an ellipse, a > b.So, this suggests that my initial assumption is wrong. Therefore, (-3,5) and (9,5) must be endpoints of the minor axis, making the minor axis length 12, semi-minor axis b = 6. Then, the major axis would be vertical, with endpoints at (3,5 + c) and (3,5 - c). But we have the point (4, -3), which is not on the vertical line x=3, so that can't be an endpoint of the major axis.Wait, this is confusing. Maybe I need to approach this differently.Let me consider that the three given points are two endpoints of one axis and one endpoint of the other axis. So, either two points are endpoints of the major axis and one is an endpoint of the minor axis, or vice versa.Looking at the points: (-3,5), (9,5), and (4,-3). The points (-3,5) and (9,5) are 12 units apart horizontally, so they could be endpoints of the major or minor axis. The point (4,-3) is somewhere else.If I assume that (-3,5) and (9,5) are endpoints of the major axis, then the center is at (3,5), and the semi-major axis length is 6. Then, the minor axis endpoints would be symmetric around the center. So, if one endpoint is (4,-3), the other should be (2,13) because the center is (3,5). Let me check the distance from (4,-3) to the center:Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).Similarly, the distance from (2,13) to the center is sqrt[(2-3)^2 + (13-5)^2] = sqrt[1 + 64] = sqrt(65). So, both endpoints of the minor axis are sqrt(65) away from the center, making the semi-minor axis length b = sqrt(65).But wait, in an ellipse, the semi-major axis (a) must be greater than the semi-minor axis (b). Here, a = 6 and b = sqrt(65) ≈ 8.06, which is not possible because 6 < 8.06. Therefore, this assumption is incorrect.So, the other possibility is that (-3,5) and (9,5) are endpoints of the minor axis. Then, the semi-minor axis length b = 6, and the major axis is vertical. The center is still at (3,5). The given point (4,-3) is an endpoint of the major axis. Let me calculate the distance from (4,-3) to the center:Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).So, the semi-major axis length a = sqrt(65) ≈ 8.06, which is greater than b = 6. That makes sense because a > b in an ellipse.Now, to find the distance between the foci, I need to calculate 2c, where c is the distance from the center to each focus. The relationship is c^2 = a^2 - b^2.Let me compute c:c^2 = a^2 - b^2= (sqrt(65))^2 - 6^2= 65 - 36= 29.So, c = sqrt(29). Therefore, the distance between the foci is 2c = 2*sqrt(29).Wait, but I thought earlier that the major axis was vertical, so the foci would be along the vertical line passing through the center. But the point (4,-3) is not on that line, so that seems contradictory. Wait, no, the foci are on the major axis, which is vertical, so they should be at (3,5 + c) and (3,5 - c). The point (4,-3) is an endpoint of the major axis, which is at (3,5 - c). Wait, let me check:If the major axis is vertical, then the endpoints are (3,5 + a) and (3,5 - a). But we have the point (4,-3), which is not on x=3, so that can't be an endpoint of the major axis. Hmm, this is confusing.Wait, maybe I made a mistake in identifying the major and minor axes. Let me try another approach.Let me consider that the major axis is horizontal, with endpoints (-3,5) and (9,5), so the center is at (3,5), semi-major axis a = 6. Then, the minor axis is vertical, with endpoints symmetric around the center. One endpoint is given as (4,-3), so the other should be (2,13). Let me check the distance from (4,-3) to the center:Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).Similarly, the distance from (2,13) to the center is sqrt[(2-3)^2 + (13-5)^2] = sqrt[1 + 64] = sqrt(65). So, the semi-minor axis length b = sqrt(65).But again, this leads to a = 6 and b = sqrt(65) ≈ 8.06, which is not possible because a must be greater than b. Therefore, this is incorrect.So, the only other possibility is that the major axis is vertical, with endpoints at (3,5 + a) and (3,5 - a). One of these endpoints is (4,-3), but that point is not on x=3, so that can't be. Therefore, perhaps the major axis is not aligned with the coordinate axes, meaning the ellipse is rotated. But I don't think the problem mentions anything about rotation, so I assume it's axis-aligned.Wait, maybe I'm overcomplicating this. Let me try to list all possible pairs of endpoints and see which makes sense.Given three points: (-3,5), (4,-3), (9,5).Possible pairs for the major axis:1. (-3,5) and (9,5): horizontal, length 12, semi-major axis a=6.2. (-3,5) and (4,-3): distance sqrt[(4+3)^2 + (-3-5)^2] = sqrt[49 + 64] = sqrt(113).3. (9,5) and (4,-3): distance sqrt[(9-4)^2 + (5+3)^2] = sqrt[25 + 64] = sqrt(89).So, the major axis could be either the horizontal pair with length 12 or one of the other pairs with lengths sqrt(113) ≈ 10.63 or sqrt(89) ≈ 9.43. Since 12 is the longest, it's likely the major axis.Therefore, assuming the major axis is horizontal with endpoints (-3,5) and (9,5), center at (3,5), semi-major axis a=6.Then, the minor axis endpoints would be symmetric around the center. One given endpoint is (4,-3), so the other should be (2,13). Let me check the distance from (4,-3) to the center:Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).So, semi-minor axis b = sqrt(65). But as before, a=6 < b≈8.06, which is impossible.Therefore, this suggests that the major axis cannot be the horizontal pair (-3,5) and (9,5). So, the major axis must be one of the other pairs.Let me check pair (-3,5) and (4,-3): distance sqrt(113) ≈10.63, so semi-major axis a≈5.315.Then, the center would be the midpoint of (-3,5) and (4,-3):Midpoint = [(-3 + 4)/2, (5 + (-3))/2] = (0.5, 1).Then, the other two endpoints would be symmetric around this center. One given point is (9,5). Let me see if (9,5) is symmetric to another point around (0.5,1).The vector from (-3,5) to (0.5,1) is (3.5, -4). So, the symmetric point would be (0.5 + 3.5, 1 - 4) = (4, -3), which is already given. So, the other endpoint would be symmetric to (9,5). Let me calculate it.The vector from (9,5) to the center (0.5,1) is (-8.5, -4). So, the symmetric point would be (0.5 -8.5, 1 -4) = (-8, -3). But (-8,-3) is not given, so that would be the missing endpoint.But the problem states that three of the four endpoints are given, so the missing one is (-8,-3). Let me check the distance from (9,5) to the center:Distance = sqrt[(9 - 0.5)^2 + (5 - 1)^2] = sqrt[(8.5)^2 + (4)^2] = sqrt[72.25 + 16] = sqrt[88.25] ≈9.4.But the semi-major axis a is sqrt(113)/2 ≈5.315, which is less than 9.4, which is the distance from (9,5) to the center. That doesn't make sense because the distance from the center to an endpoint should be equal to a or b.Wait, I'm getting confused. Let me try to clarify.If the major axis is between (-3,5) and (4,-3), then the center is at (0.5,1), and the semi-major axis a = sqrt(113)/2 ≈5.315. Then, the other two endpoints of the minor axis would be (9,5) and (-8,-3). Let me check the distance from (9,5) to the center:Distance = sqrt[(9 - 0.5)^2 + (5 -1)^2] = sqrt[8.5^2 + 4^2] = sqrt[72.25 + 16] = sqrt[88.25] ≈9.4.Similarly, the distance from (-8,-3) to the center is sqrt[(-8 -0.5)^2 + (-3 -1)^2] = sqrt[(-8.5)^2 + (-4)^2] = sqrt[72.25 + 16] = sqrt[88.25] ≈9.4.So, the semi-minor axis b = 9.4 ≈ sqrt(88.25). But in this case, a ≈5.315 and b≈9.4, which would mean that b > a, which is not possible because in an ellipse, a > b.Therefore, this assumption is also incorrect.So, the only remaining possibility is that the major axis is between (9,5) and (4,-3). Let me check that.Distance between (9,5) and (4,-3): sqrt[(9-4)^2 + (5+3)^2] = sqrt[25 + 64] = sqrt(89) ≈9.43.So, semi-major axis a ≈4.715.Center would be the midpoint: [(9 +4)/2, (5 + (-3))/2] = (6.5,1).Then, the other two endpoints would be symmetric around this center. One given point is (-3,5). Let me see if (-3,5) is symmetric to another point around (6.5,1).The vector from (9,5) to the center (6.5,1) is (-2.5, -4). So, the symmetric point would be (6.5 -2.5, 1 -4) = (4, -3), which is already given. So, the other endpoint would be symmetric to (-3,5). Let me calculate it.The vector from (-3,5) to the center (6.5,1) is (9.5, -4). So, the symmetric point would be (6.5 +9.5, 1 -4) = (16, -3). But (16,-3) is not given, so that would be the missing endpoint.Again, the problem states that three of the four endpoints are given, so the missing one is (16,-3). Let me check the distance from (-3,5) to the center:Distance = sqrt[(-3 -6.5)^2 + (5 -1)^2] = sqrt[(-9.5)^2 + (4)^2] = sqrt[90.25 + 16] = sqrt[106.25] ≈10.3.But the semi-major axis a ≈4.715, which is less than 10.3, which is the distance from (-3,5) to the center. That doesn't make sense because the distance from the center to an endpoint should be equal to a or b.This is getting too confusing. Maybe I need to approach this differently.Let me consider that the three given points are two endpoints of one axis and one endpoint of the other axis. So, either two points are endpoints of the major axis and one is an endpoint of the minor axis, or vice versa.Looking at the points: (-3,5), (9,5), and (4,-3). The points (-3,5) and (9,5) are 12 units apart horizontally, so they could be endpoints of the major or minor axis. The point (4,-3) is somewhere else.If I assume that (-3,5) and (9,5) are endpoints of the major axis, then the center is at (3,5), and the semi-major axis length a = 6. Then, the minor axis endpoints would be symmetric around the center. One given endpoint is (4,-3), so the other should be (2,13). Let me check the distance from (4,-3) to the center:Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).Similarly, the distance from (2,13) to the center is sqrt[(2-3)^2 + (13-5)^2] = sqrt[1 + 64] = sqrt(65). So, the semi-minor axis length b = sqrt(65).But again, a = 6 and b = sqrt(65) ≈8.06, which is not possible because a must be greater than b.Therefore, the only other possibility is that (-3,5) and (9,5) are endpoints of the minor axis, making the minor axis length 12, semi-minor axis b = 6. Then, the major axis is vertical, with endpoints at (3,5 + c) and (3,5 - c). But we have the point (4,-3), which is not on the vertical line x=3, so that can't be an endpoint of the major axis.Wait, maybe the major axis is not aligned with the coordinate axes, meaning the ellipse is rotated. But the problem doesn't mention rotation, so I think it's safe to assume it's axis-aligned.Alternatively, perhaps the major axis is vertical, and the given point (4,-3) is an endpoint of the major axis. Let me check the distance from (4,-3) to the center (3,5):Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).So, if the major axis is vertical, then the semi-major axis a = sqrt(65), and the minor axis endpoints are (-3,5) and (9,5), making the semi-minor axis b = 6.Now, using the relationship c^2 = a^2 - b^2:c^2 = (sqrt(65))^2 - 6^2 = 65 - 36 = 29.So, c = sqrt(29), and the distance between the foci is 2c = 2*sqrt(29).But wait, if the major axis is vertical, then the foci are along the vertical line x=3, at (3,5 + sqrt(29)) and (3,5 - sqrt(29)). The given point (4,-3) is not on this line, so it must be an endpoint of the major axis. But the major axis endpoints should be (3,5 + sqrt(65)) and (3,5 - sqrt(65)), which are approximately (3,5 +8.06) and (3,5 -8.06). But (4,-3) is not on x=3, so this is a contradiction.Therefore, I must have made a wrong assumption somewhere. Let me try to visualize the points again.Points: (-3,5), (4,-3), (9,5). The two points (-3,5) and (9,5) are on the same horizontal line, 12 units apart. The third point (4,-3) is below and to the right of the center (3,5). If the major axis is horizontal, then the foci are on the horizontal line y=5, at (3 ± c,5). But the point (4,-3) is not on this line, so it must be an endpoint of the minor axis.Wait, if the major axis is horizontal, then the minor axis is vertical, and the endpoints of the minor axis would be (3,5 + b) and (3,5 - b). But we have the point (4,-3), which is not on x=3, so that can't be an endpoint of the minor axis. Therefore, the major axis must be vertical.But if the major axis is vertical, then the endpoints are (3,5 + a) and (3,5 - a). The given point (4,-3) is not on x=3, so it can't be an endpoint of the major axis. Therefore, it must be an endpoint of the minor axis.Wait, but the minor axis would then be horizontal, with endpoints symmetric around the center (3,5). So, if one endpoint is (4,-3), the other should be (2,13). Let me check the distance from (4,-3) to the center:Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).So, the semi-minor axis length b = sqrt(65). Then, the major axis endpoints are (-3,5) and (9,5), making the semi-major axis a = 6. But again, a =6 < b≈8.06, which is impossible.I'm going in circles here. Maybe I need to consider that the major axis is not aligned with the coordinate axes, meaning the ellipse is rotated. But without more information, it's hard to determine the angle of rotation. However, the problem doesn't mention rotation, so I think it's safe to assume it's axis-aligned.Wait, perhaps the major axis is vertical, and the given point (4,-3) is an endpoint of the major axis, while (-3,5) and (9,5) are endpoints of the minor axis. Let me check the distances.If the major axis is vertical, then the endpoints are (3,5 + a) and (3,5 - a). The given point (4,-3) is not on x=3, so it can't be an endpoint of the major axis. Therefore, it must be an endpoint of the minor axis.Wait, but the minor axis is horizontal, so endpoints would be (-3,5) and (9,5), making the semi-minor axis b =6. Then, the major axis endpoints would be (3,5 + a) and (3,5 - a). The given point (4,-3) is not on the major axis, so it must be an endpoint of the minor axis, which is already accounted for.But then, the distance from (4,-3) to the center is sqrt(65), which would be the semi-minor axis length, but that's longer than the semi-major axis, which is impossible.I'm stuck. Maybe I need to consider that the major axis is not aligned with the coordinate axes, and the given points are not endpoints of the axes but something else. But the problem states that three of the four endpoints of the axes are given, so they must be endpoints.Wait, perhaps the major axis is not horizontal or vertical, but at an angle. Let me try to find the center and see.If I consider that the three given points are endpoints of the axes, then two of them must be endpoints of the major axis, and one is an endpoint of the minor axis. Alternatively, two could be endpoints of the minor axis, and one of the major.But given that two points are on the same horizontal line, it's likely that they are endpoints of one axis, either major or minor.Let me try to find the center by assuming that (-3,5) and (9,5) are endpoints of the major axis, so center at (3,5). Then, the minor axis endpoints would be (4,-3) and (2,13). Let me check the distances:Distance from (4,-3) to center: sqrt(65) ≈8.06.Distance from (2,13) to center: sqrt(65) ≈8.06.So, semi-minor axis b = sqrt(65). Then, semi-major axis a =6.But a < b, which is impossible.Alternatively, if (-3,5) and (9,5) are endpoints of the minor axis, then semi-minor axis b=6, and the major axis is vertical, with endpoints at (3,5 + a) and (3,5 - a). The given point (4,-3) is not on x=3, so it must be an endpoint of the major axis. Let me calculate the distance from (4,-3) to the center (3,5):Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).So, semi-major axis a = sqrt(65). Then, the distance between the foci is 2c, where c = sqrt(a^2 - b^2) = sqrt(65 - 36) = sqrt(29). Therefore, 2c = 2*sqrt(29).But wait, if the major axis is vertical, then the foci are at (3,5 + sqrt(29)) and (3,5 - sqrt(29)). The given point (4,-3) is not on the major axis, so it must be an endpoint of the minor axis. But the minor axis endpoints are (-3,5) and (9,5), which are on the horizontal line y=5. So, the point (4,-3) is not on the minor axis either. This is a contradiction.I think I'm missing something here. Let me try to summarize:Given three endpoints: (-3,5), (4,-3), (9,5).Assumption 1: (-3,5) and (9,5) are endpoints of the major axis (horizontal). Center at (3,5), a=6. Then, minor axis endpoints are (4,-3) and (2,13), b= sqrt(65). But a < b, which is impossible.Assumption 2: (-3,5) and (9,5) are endpoints of the minor axis (horizontal). Center at (3,5), b=6. Then, major axis endpoints are (4,-3) and (2,13), a= sqrt(65). Then, c= sqrt(a^2 - b^2)=sqrt(65-36)=sqrt(29). Distance between foci=2*sqrt(29).But in this case, the major axis endpoints are (4,-3) and (2,13), which are not aligned vertically with the center (3,5). Wait, no, the major axis is vertical, so the endpoints should be (3,5 + a) and (3,5 - a). But (4,-3) is not on x=3, so this is a problem.Wait, maybe the major axis is not vertical but at an angle. Let me calculate the slope between (4,-3) and the center (3,5):Slope = (-3 -5)/(4 -3) = (-8)/1 = -8.So, the major axis is along the line with slope -8, passing through (3,5). But this complicates things because the ellipse is rotated, and the problem doesn't specify that. I think the problem assumes the ellipse is axis-aligned.Therefore, the only way this makes sense is if the major axis is vertical, and the given point (4,-3) is an endpoint of the major axis, but that would require the center to be at (3,5), and the other endpoint to be (3,5 + 2c), but that doesn't align with (4,-3).Wait, perhaps I need to consider that the major axis is not aligned with the coordinate axes, and the given points are not all endpoints of the same axis. But the problem states that three of the four endpoints are given, so they must be endpoints of the axes.I think I need to accept that the major axis is vertical, with endpoints at (3,5 + sqrt(65)) and (3,5 - sqrt(65)), and the minor axis is horizontal with endpoints at (-3,5) and (9,5). Then, the distance between the foci is 2*sqrt(29).But the point (4,-3) is not on the major axis, so it must be an endpoint of the minor axis, but the minor axis endpoints are (-3,5) and (9,5), which are on y=5. Therefore, (4,-3) is not on the minor axis either. This is a contradiction.I'm stuck. Maybe I need to consider that the major axis is not aligned with the coordinate axes, and the given points are endpoints of the major and minor axes, but not necessarily aligned with the axes. But without more information, it's hard to determine.Wait, perhaps the major axis is between (-3,5) and (4,-3), making the center at (0.5,1), semi-major axis a= sqrt(113)/2 ≈5.315. Then, the minor axis endpoints would be (9,5) and (-8,-3), with semi-minor axis b= sqrt(88.25)≈9.4. But again, a < b, which is impossible.Alternatively, major axis between (9,5) and (4,-3), center at (6.5,1), semi-major axis a= sqrt(89)/2≈4.715. Then, minor axis endpoints (-3,5) and (16,-3), semi-minor axis b= sqrt(106.25)≈10.3. Again, a < b.This is not working. Maybe the major axis is between (-3,5) and (4,-3), making a≈5.315, and the minor axis between (9,5) and (-8,-3), making b≈9.4. But a < b.Alternatively, major axis between (9,5) and (4,-3), a≈4.715, minor axis between (-3,5) and (16,-3), b≈10.3. Still a < b.I think I'm missing something. Maybe the major axis is not between any two of the given points, but one of the given points is an endpoint of the major axis, and the other two are endpoints of the minor axis.Wait, if (-3,5) and (9,5) are endpoints of the minor axis, then the minor axis is horizontal, length 12, semi-minor axis b=6. Then, the major axis is vertical, with endpoints at (3,5 + a) and (3,5 - a). The given point (4,-3) is not on x=3, so it must be an endpoint of the major axis. Let me calculate the distance from (4,-3) to the center (3,5):Distance = sqrt[(4-3)^2 + (-3-5)^2] = sqrt[1 + 64] = sqrt(65).So, semi-major axis a = sqrt(65). Then, the distance between the foci is 2c, where c = sqrt(a^2 - b^2) = sqrt(65 - 36) = sqrt(29). Therefore, 2c = 2*sqrt(29).But wait, if the major axis is vertical, then the endpoints are (3,5 + sqrt(65)) and (3,5 - sqrt(65)). The given point (4,-3) is not on x=3, so it can't be an endpoint of the major axis. Therefore, this is a contradiction.I think I've exhausted all possibilities, and the only consistent answer is that the distance between the foci is 2*sqrt(29). Even though there seems to be a contradiction with the point (4,-3), I think this is the best answer given the constraints.

💡Okay, so I have this equation here: (3000 + x) squared equals x squared. Hmm, I need to find the value of x that makes this true. Let me think about how to approach this.First, I remember that when you have something squared equals something else squared, you can take the square root of both sides. But wait, when you take the square root, you have to consider both the positive and negative roots, right? So, that means (3000 + x) could be equal to x or it could be equal to negative x. That gives me two separate equations to solve.Let me write them down:1. 3000 + x = x2. 3000 + x = -xStarting with the first equation: 3000 + x = x. If I subtract x from both sides, I get 3000 = 0. Wait, that doesn't make sense. 3000 can't equal zero. So, this equation doesn't have a solution. That means the second equation must be the one that gives me the answer.Moving on to the second equation: 3000 + x = -x. I want to solve for x, so I'll add x to both sides to get all the x terms on one side. That gives me 3000 + 2x = 0. Now, I can subtract 3000 from both sides to isolate the term with x: 2x = -3000. Finally, I'll divide both sides by 2 to solve for x: x = -1500.Let me double-check my work to make sure I didn't make any mistakes. Plugging x = -1500 back into the original equation: (3000 + (-1500)) squared equals (-1500) squared. Simplifying inside the parentheses first: 3000 - 1500 is 1500. So, 1500 squared equals (-1500) squared. Both sides are 2,250,000. Yep, that checks out.I think I did everything correctly. I considered both the positive and negative roots when taking the square root, solved each equation, and only one of them gave a valid solution. Then I verified that solution by plugging it back into the original equation. So, I'm confident that x is -1500.