💡Okay, so I have this quadratic equation: x squared minus 8x plus 18. I need to find the value of x that gives the minimum value for this quadratic. Hmm, quadratics are parabolas, right? And since the coefficient of x squared is positive (which is 1 in this case), the parabola opens upwards. That means it has a minimum point, which is the vertex of the parabola. So, I need to find the vertex of this quadratic to determine the x-value that gives the minimum value.I remember there are a couple of ways to find the vertex of a quadratic. One way is by completing the square, and another is using the formula for the vertex. I think the formula is x equals negative b over 2a, where a and b are the coefficients from the quadratic equation in standard form ax squared plus bx plus c. Let me try that first because it seems straightforward.In my equation, x squared minus 8x plus 18, the coefficients are a = 1, b = -8, and c = 18. So, applying the formula, x equals negative b over 2a would be x equals negative (-8) over 2 times 1. That simplifies to x equals 8 over 2, which is x equals 4. So, according to this formula, the x-value that gives the minimum is 4.Wait, but I should probably double-check this by completing the square to make sure I didn't make a mistake. Let me try that method as well. Completing the square involves rewriting the quadratic in the form (x - h)^2 plus k, where (h, k) is the vertex of the parabola.Starting with the quadratic: x squared minus 8x plus 18. I need to complete the square for the x terms. So, I take the coefficient of x, which is -8, divide it by 2 to get -4, and then square it to get 16. So, I can rewrite the quadratic as (x - 4)^2 minus 16 plus 18. That simplifies to (x - 4)^2 plus 2. Now, in this form, the vertex is at (4, 2). Since the square term (x - 4)^2 is always non-negative, the smallest value it can take is 0, which happens when x equals 4. Therefore, the minimum value of the quadratic is 2, and it occurs at x = 4. So, both methods give me the same result: x equals 4. That makes me confident that this is the correct answer. I think I've got it right. The value of x that gives the minimum value for the quadratic x squared minus 8x plus 18 is 4.Just to be thorough, maybe I can plug x = 4 back into the original equation to verify. Let's see: (4)^2 minus 8*(4) plus 18. That's 16 minus 32 plus 18. 16 minus 32 is -16, and -16 plus 18 is 2. Yep, that matches the minimum value I found earlier. So, everything checks out.I guess another way to think about it is graphing the quadratic. If I plot y = x squared minus 8x plus 18, it should be a parabola opening upwards with its vertex at (4, 2). So, the lowest point on the graph is at x = 4, which confirms that this is indeed the x-value where the minimum occurs.I wonder if there's a calculus way to find this minimum. Maybe taking the derivative and setting it equal to zero? Let me try that real quick. The derivative of y with respect to x is 2x minus 8. Setting that equal to zero gives 2x minus 8 equals 0, so 2x equals 8, which means x equals 4. Again, the same result. That's reassuring.So, whether I use algebraic methods like the vertex formula or completing the square, or calculus by taking the derivative, I consistently get x = 4 as the point where the quadratic reaches its minimum value. I think I'm confident enough now that this is the correct answer.Just to recap, the steps I took were:1. Recognize that the quadratic opens upwards, so it has a minimum.2. Use the vertex formula x = -b/(2a) to find the x-value of the vertex.3. Verified the result by completing the square.4. Plugged the x-value back into the original equation to confirm the minimum value.5. Cross-checked using calculus by finding the derivative and solving for x.All these methods led me to the same conclusion, so I'm pretty sure that x = 4 is the correct answer. It's always good to verify using multiple methods to ensure accuracy, especially when dealing with mathematical problems where a small mistake can lead to a wrong answer.I guess another thing to consider is whether there are any constraints or special cases I need to be aware of. In this problem, since it's a standard quadratic with real coefficients, there shouldn't be any issues. The quadratic is defined for all real numbers, so the vertex is indeed the point of minimum value.I also remember that the vertex form of a quadratic is useful because it directly gives the vertex coordinates. In this case, after completing the square, I had (x - 4)^2 + 2, which clearly shows the vertex at (4, 2). This form is handy because it eliminates any ambiguity about where the vertex is located.Thinking about the graph again, since the parabola opens upwards, the arms of the parabola go upwards on both sides of the vertex. This means that as x moves away from 4 in either direction, the value of the quadratic increases. So, x = 4 is the point where the quadratic is at its lowest, confirming that it's the minimum.I wonder if there's a geometric interpretation of this. Maybe if I think of the quadratic as modeling some real-world phenomenon, like the height of a projectile over time, the vertex would represent the maximum height. But in this case, since it's a minimum, perhaps it's modeling something like the distance from a point, where the minimum distance occurs at x = 4.Anyway, regardless of the application, the mathematical process remains the same. The key takeaway is that for any quadratic equation in standard form, the vertex can be found using the formula x = -b/(2a), and this vertex represents either the minimum or maximum value of the quadratic depending on the direction it opens.In summary, after carefully considering multiple methods and verifying each step, I'm confident that the value of x that gives the minimum value for the quadratic x squared minus 8x plus 18 is indeed 4.

💡First, calculate the relative speed between the train and the man by adding their speeds since they are moving in opposite directions.Relative speed = Speed of train + Speed of manRelative speed = 84.99280057595394 km/h + 5 km/hRelative speed = 89.99280057595394 km/hNext, convert the relative speed from kilometers per hour to meters per second.Relative speed in m/s = 89.99280057595394 km/h × (1000 m / 3600 s)Relative speed in m/s = 24.99800015998776 m/sFinally, use the relative speed and the time taken to cross the man to find the length of the train.Length of train = Relative speed × TimeLength of train = 24.99800015998776 m/s × 6 sLength of train ≈ 149.99 meters

💡Okay, so I'm trying to figure out how to convince my beloved that I'm a rich liar using just one phrase. Let's see... I remember that on this island, there are knights who always tell the truth and liars who always lie. My beloved doesn't want to marry a knight; she only wants to marry a liar, and specifically a rich one. I happen to be a rich liar, so I need to find a way to communicate that to her with just one statement.First, I need to think about what a knight would say versus what a liar would say. Knights always tell the truth, so if I were a knight, I would have to say something truthful. But I'm not a knight; I'm a liar. So whatever I say must be a lie. But I want her to understand that I'm a rich liar, not just any liar.Maybe I can start by considering what a liar would say if they were poor. If I were a poor liar, I might say something like, "I am a poor liar." But wait, if I say that, and I'm actually a poor liar, then that statement would be true, which contradicts the fact that I'm a liar who always lies. So that can't be right.Hmm, maybe I need to think about it differently. If I say, "I am a poor liar," and I'm actually a rich liar, then that statement would be false, which fits because I'm a liar. So by saying "I am a poor liar," I'm lying, which tells her that I'm not a poor liar, meaning I must be a rich liar. That seems to make sense.But let me double-check. If I say, "I am a poor liar," and I'm a liar, then the statement must be false. Therefore, I am not a poor liar, which means I must be a rich liar. Yes, that works. So by making that statement, I'm effectively telling her that I'm a rich liar without directly saying it.Wait, but what if she doesn't understand the logic behind it? Maybe I need to make it clearer. Is there another way to phrase it? What if I say, "I am not a rich liar." If I'm a rich liar, then saying "I am not a rich liar" would be a lie, which fits. But then she might think I'm a poor liar, which is not what I want.No, that doesn't help. The first idea seems better. By saying "I am a poor liar," I'm lying, which tells her I'm not a poor liar, hence I must be a rich liar. That seems to be the most straightforward way to communicate my status as a rich liar in one statement.I think that's the solution. I'll go with that.

💡First, I need to find the value of ( t ) when ( x = -6 ) using the equation ( x = 3 - 2t ).Substituting ( x = -6 ) into the equation:[-6 = 3 - 2t]Next, I'll solve for ( t ):[-6 - 3 = -2t -9 = -2t t = frac{-9}{-2} t = 4.5]Now that I have ( t = 4.5 ), I'll substitute this value into the equation for ( y ) to find ( y ):[y = 3t + 6 y = 3(4.5) + 6 y = 13.5 + 6 y = 19.5]Therefore, the value of ( y ) when ( x = -6 ) is ( 19.5 ).

💡Okay, so I have this problem where ( x ) is defined as a continued fraction: ( x = 1 + frac{sqrt{3}}{1 + frac{sqrt{3}}{1 + dots}} ). I need to find the value of ( frac{1}{(x+2)(x-3)} ) and express it in the form ( frac{A+sqrt{B}}{C} ), where ( A ), ( B ), and ( C ) are integers, and ( B ) isn't divisible by the square of a prime. Then, I have to find the sum of the absolute values of ( A ), ( B ), and ( C ).Alright, let me start by trying to understand the continued fraction. It looks like ( x ) is defined recursively. That is, the expression for ( x ) contains itself within it. So, maybe I can set up an equation based on that.Let me write down the definition again:( x = 1 + frac{sqrt{3}}{1 + frac{sqrt{3}}{1 + dots}} )Notice that the part after the first "1 +" is similar to the entire expression for ( x ). So, if I let ( x ) be equal to that entire continued fraction, then the denominator inside the fraction is also ( x ). That seems right.So, I can write:( x = 1 + frac{sqrt{3}}{x} )Wait, is that correct? Let me check. The continued fraction is ( 1 + frac{sqrt{3}}{1 + frac{sqrt{3}}{1 + dots}} ), so yes, the denominator is the same as the entire expression, which is ( x ). So, that equation should hold.So, now I have:( x = 1 + frac{sqrt{3}}{x} )I can rearrange this equation to solve for ( x ). Let me do that.First, multiply both sides by ( x ) to eliminate the denominator:( x cdot x = x cdot 1 + sqrt{3} )Which simplifies to:( x^2 = x + sqrt{3} )Then, bring all terms to one side:( x^2 - x - sqrt{3} = 0 )So, that's a quadratic equation in terms of ( x ). Hmm, interesting. I can use the quadratic formula to solve for ( x ), but maybe I don't need to find the exact value of ( x ) right away. Let me see what the problem is asking for.The problem wants me to find ( frac{1}{(x+2)(x-3)} ). Let me compute the denominator first:( (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 )So, ( frac{1}{(x+2)(x-3)} = frac{1}{x^2 - x - 6} )But from the equation I derived earlier, ( x^2 - x = sqrt{3} ). So, substituting that into the denominator:( x^2 - x - 6 = sqrt{3} - 6 )Therefore, the expression simplifies to:( frac{1}{sqrt{3} - 6} )Now, I need to rationalize the denominator because it has a radical. To do that, I'll multiply both the numerator and the denominator by the conjugate of the denominator, which is ( sqrt{3} + 6 ):( frac{1}{sqrt{3} - 6} times frac{sqrt{3} + 6}{sqrt{3} + 6} = frac{sqrt{3} + 6}{(sqrt{3})^2 - (6)^2} )Calculating the denominator:( (sqrt{3})^2 = 3 ) and ( 6^2 = 36 ), so:( 3 - 36 = -33 )Therefore, the expression becomes:( frac{sqrt{3} + 6}{-33} )I can write this as:( frac{6 + sqrt{3}}{-33} )But usually, we prefer to have the negative sign in the numerator or the denominator. Let me write it as:( frac{6 + sqrt{3}}{-33} = -frac{6 + sqrt{3}}{33} )However, the problem specifies the form ( frac{A + sqrt{B}}{C} ), where ( A ), ( B ), and ( C ) are integers, and ( B ) is not divisible by the square of a prime. So, I need to make sure that the denominator is positive. Therefore, I can write:( frac{6 + sqrt{3}}{-33} = frac{-6 - sqrt{3}}{33} )But in this case, ( A ) would be -6, ( B ) would be 3, and ( C ) would be 33. However, the problem doesn't specify that ( A ) has to be positive, just that ( A ), ( B ), and ( C ) are integers. So, I think both forms are acceptable, but let me check if I can write it with positive ( A ).Alternatively, I can factor out the negative sign:( frac{6 + sqrt{3}}{-33} = -frac{6 + sqrt{3}}{33} )But in this case, the negative sign is outside the fraction, which might not fit the form they want. So, perhaps it's better to write it as:( frac{-6 - sqrt{3}}{33} )So, ( A = -6 ), ( B = 3 ), and ( C = 33 ). Alternatively, if I prefer positive ( A ), I can write it as:( frac{6 + sqrt{3}}{-33} )But then ( C ) is negative. The problem doesn't specify whether ( C ) has to be positive, but it's more conventional to have the denominator positive. So, I think the correct form is:( frac{-6 - sqrt{3}}{33} )But let me double-check my steps to make sure I didn't make a mistake.Starting from the beginning:1. Defined ( x = 1 + frac{sqrt{3}}{x} )2. Multiplied both sides by ( x ): ( x^2 = x + sqrt{3} )3. Rearranged: ( x^2 - x - sqrt{3} = 0 )4. Then, ( (x + 2)(x - 3) = x^2 - x - 6 )5. Substituted ( x^2 - x = sqrt{3} ): ( sqrt{3} - 6 )6. So, ( frac{1}{sqrt{3} - 6} )7. Rationalized: ( frac{sqrt{3} + 6}{-33} )Yes, that seems correct. So, the expression is ( frac{sqrt{3} + 6}{-33} ), which can be written as ( frac{6 + sqrt{3}}{-33} ) or ( frac{-6 - sqrt{3}}{33} ).Since the problem asks for ( A ), ( B ), and ( C ) as integers, and ( B ) not divisible by the square of a prime, both forms are acceptable, but I think the second form with positive denominator is preferable.So, ( A = -6 ), ( B = 3 ), ( C = 33 ). Therefore, ( |A| + |B| + |C| = | -6 | + |3| + |33| = 6 + 3 + 33 = 42 ).Wait, but in my initial calculation, I wrote ( frac{6 + sqrt{3}}{-33} ), which would be ( A = 6 ), ( B = 3 ), ( C = -33 ). Then, ( |A| + |B| + |C| = 6 + 3 + 33 = 42 ) as well.So, regardless of how I write it, the sum is 42. Therefore, the final answer is 42.But let me just make sure I didn't make any mistakes in the rationalization step.Starting from ( frac{1}{sqrt{3} - 6} ), multiplying numerator and denominator by ( sqrt{3} + 6 ):Numerator: ( 1 times (sqrt{3} + 6) = sqrt{3} + 6 )Denominator: ( (sqrt{3} - 6)(sqrt{3} + 6) = (sqrt{3})^2 - (6)^2 = 3 - 36 = -33 )Yes, that's correct. So, the result is ( frac{sqrt{3} + 6}{-33} ), which is the same as ( frac{6 + sqrt{3}}{-33} ) or ( -frac{6 + sqrt{3}}{33} ).Since the problem doesn't specify that ( A ), ( B ), or ( C ) must be positive, just that they are integers, and ( B ) is square-free, both forms are acceptable. However, to have ( C ) positive, I think the form ( frac{-6 - sqrt{3}}{33} ) is better, but in terms of absolute values, it doesn't matter because we take the absolute value anyway.So, whether ( A ) is 6 or -6, the absolute value is 6. Similarly, ( C ) is 33 or -33, absolute value is 33. And ( B ) is 3 in both cases. So, the sum is 6 + 3 + 33 = 42.Therefore, I'm confident that the answer is 42.

💡First, I note that the bus initially has 15 passengers.Next, I will process each of the four stops one by one, updating the number of passengers after each stop.At the first stop, 3 passengers board the bus and 6 passengers alight. This results in a net change of -3 passengers, leaving 12 passengers on the bus.At the second stop, 2 passengers alight and 4 passengers board. This results in a net change of +2 passengers, increasing the total to 14 passengers.At the third stop, 7 passengers alight and 2 passengers board. This results in a net change of -5 passengers, reducing the total to 9 passengers.Finally, at the fourth stop, 3 passengers board and 5 passengers alight. This results in a net change of -2 passengers, leaving 7 passengers on the bus.Therefore, after passing through all four stops, there are 7 passengers on the bus.