💡First, I need to determine the number of employees participating in the pension program from each shift.For the first shift, there are 60 employees, and 20% of them participate in the pension program. So, 20% of 60 is 12 employees.For the second shift, there are 50 employees, and 40% of them participate in the pension program. So, 40% of 50 is 20 employees.Next, I need to find out how many employees from the third shift participate in the pension program. Let's denote this number as P.The total number of employees in the company is the sum of all three shifts: 60 + 50 + 40, which equals 150 employees.According to the problem, 24% of all employees participate in the pension program. So, 24% of 150 employees is 36 employees.Now, I can set up an equation to find P:12 (from the first shift) + 20 (from the second shift) + P (from the third shift) = 36Solving for P:12 + 20 + P = 3632 + P = 36P = 36 - 32P = 4So, 4 employees from the third shift participate in the pension program.Finally, to find the percentage of the third shift that participates in the pension program, I divide the number of participants by the total number of employees in the third shift and multiply by 100:(4 / 40) * 100 = 10%Therefore, 10% of the third shift participates in the pension program.

💡Okay, so I have this problem about a hyperbola, and I need to find its eccentricity. Let me try to understand what's given and what I need to do step by step.First, the hyperbola is given by the equation (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), where (a > 0) and (b > 0). I know that for hyperbolas, the standard form is similar to this, and the foci are located at ((pm c, 0)), where (c^2 = a^2 + b^2). The eccentricity (e) is defined as (e = frac{c}{a}), so that's what I need to find in the end.Now, the problem says that a line with slope 1 passes through the left focus of the hyperbola and intersects the curve at points (A) and (B). The left focus is at ((-c, 0)), right? So the equation of the line with slope 1 passing through ((-c, 0)) should be (y = x + c). Let me write that down.Next, this line intersects the hyperbola at points (A) and (B). So I need to find the coordinates of these intersection points. To do that, I can substitute (y = x + c) into the hyperbola equation.Substituting (y) into the hyperbola equation:[frac{x^2}{a^2} - frac{(x + c)^2}{b^2} = 1]Let me expand this equation:[frac{x^2}{a^2} - frac{x^2 + 2cx + c^2}{b^2} = 1]To combine these terms, I'll get a common denominator, which is (a^2b^2). Multiplying each term accordingly:[frac{b^2x^2 - a^2x^2 - 2a^2cx - a^2c^2}{a^2b^2} = 1]Simplifying the numerator:[(b^2 - a^2)x^2 - 2a^2cx - a^2c^2 = a^2b^2]Wait, actually, I think I missed a term when moving the 1 to the left side. Let me correct that. The equation should be:[frac{x^2}{a^2} - frac{(x + c)^2}{b^2} - 1 = 0]So, multiplying through by (a^2b^2):[b^2x^2 - a^2(x^2 + 2cx + c^2) - a^2b^2 = 0]Expanding the terms:[b^2x^2 - a^2x^2 - 2a^2cx - a^2c^2 - a^2b^2 = 0]Combining like terms:[(b^2 - a^2)x^2 - 2a^2cx - (a^2c^2 + a^2b^2) = 0]So, that's a quadratic equation in terms of (x). Let me denote this as:[(b^2 - a^2)x^2 - 2a^2cx - a^2(c^2 + b^2) = 0]Let me write this as:[(b^2 - a^2)x^2 - 2a^2cx - a^2(c^2 + b^2) = 0]This quadratic equation will have two solutions, corresponding to the (x)-coordinates of points (A) and (B). Let me denote the roots as (x_1) and (x_2). From quadratic equation properties, the sum of the roots (x_1 + x_2) is equal to (frac{2a^2c}{b^2 - a^2}), and the product (x_1x_2) is (frac{-a^2(c^2 + b^2)}{b^2 - a^2}).But wait, the problem mentions the vector sum (overrightarrow{OA} + overrightarrow{OB}) being collinear with the vector (overrightarrow{n} = (-3, -1)). So, the vector sum is ((x_1 + x_2, y_1 + y_2)), and this should be a scalar multiple of ((-3, -1)). That means the coordinates of the vector sum must satisfy the ratio (frac{y_1 + y_2}{x_1 + x_2} = frac{-1}{-3} = frac{1}{3}).So, I can write:[frac{y_1 + y_2}{x_1 + x_2} = frac{1}{3}]But since the points (A) and (B) lie on the line (y = x + c), their (y)-coordinates are (y_1 = x_1 + c) and (y_2 = x_2 + c). Therefore, the sum (y_1 + y_2 = x_1 + x_2 + 2c).Substituting this into the ratio:[frac{x_1 + x_2 + 2c}{x_1 + x_2} = frac{1}{3}]Let me denote (S = x_1 + x_2 = frac{2a^2c}{b^2 - a^2}). Then the equation becomes:[frac{S + 2c}{S} = frac{1}{3}]Simplifying:[1 + frac{2c}{S} = frac{1}{3}]Subtracting 1 from both sides:[frac{2c}{S} = -frac{2}{3}]Multiplying both sides by (S):[2c = -frac{2}{3}S]Substituting (S = frac{2a^2c}{b^2 - a^2}):[2c = -frac{2}{3} cdot frac{2a^2c}{b^2 - a^2}]Simplify the right side:[2c = -frac{4a^2c}{3(b^2 - a^2)}]Divide both sides by (2c) (assuming (c neq 0), which it isn't because it's a hyperbola):[1 = -frac{2a^2}{3(b^2 - a^2)}]Multiply both sides by (3(b^2 - a^2)):[3(b^2 - a^2) = -2a^2]Expanding:[3b^2 - 3a^2 = -2a^2]Bring all terms to one side:[3b^2 - 3a^2 + 2a^2 = 0]Simplify:[3b^2 - a^2 = 0]So,[3b^2 = a^2]Which implies:[a^2 = 3b^2]So, now I have a relationship between (a) and (b). Remember, for hyperbolas, (c^2 = a^2 + b^2). Let's substitute (a^2 = 3b^2) into this:[c^2 = 3b^2 + b^2 = 4b^2]Therefore,[c = 2b]But we need the eccentricity (e = frac{c}{a}). Let's express (c) and (a) in terms of (b):[c = 2b, quad a = sqrt{3}b]So,[e = frac{c}{a} = frac{2b}{sqrt{3}b} = frac{2}{sqrt{3}} = frac{2sqrt{3}}{3}]So, the eccentricity is (frac{2sqrt{3}}{3}), which corresponds to option B.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Equation of the line: (y = x + c) – correct.2. Substituted into hyperbola: Correctly expanded and simplified to a quadratic in (x).3. Sum of roots (x_1 + x_2 = frac{2a^2c}{b^2 - a^2}) – correct.4. Sum of (y)-coordinates: (y_1 + y_2 = x_1 + x_2 + 2c) – correct.5. Set up the ratio (frac{y_1 + y_2}{x_1 + x_2} = frac{1}{3}) – correct.6. Solved for (S) and found (a^2 = 3b^2) – correct.7. Calculated (c^2 = 4b^2) so (c = 2b) – correct.8. Eccentricity (e = frac{2}{sqrt{3}} = frac{2sqrt{3}}{3}) – correct.Everything seems to check out. So, I think the answer is indeed option B.**Final Answer**boxed{B}

💡To determine how the graph of ( y = 3sinleft(2x + frac{pi}{4}right) ) is obtained from the graph of ( y = 3sin(2x) ), we analyze the transformation applied to the function.1. **Start with the original function:** [ y = 3sin(2x) ]2. **Express the transformed function:** [ y = 3sinleft(2x + frac{pi}{4}right) ]3. **Factor out the coefficient of ( x ) inside the sine function:** [ y = 3sinleft(2left(x + frac{pi}{8}right)right) ] This is done by rewriting ( 2x + frac{pi}{4} ) as ( 2left(x + frac{pi}{8}right) ).4. **Interpret the transformation:** The expression ( 2left(x + frac{pi}{8}right) ) indicates a horizontal shift. Specifically, adding ( frac{pi}{8} ) inside the function argument results in a shift to the left by ( frac{pi}{8} ) units.5. **Conclusion:** The graph of ( y = 3sin(2x) ) must be shifted ( frac{pi}{8} ) units to the left to obtain the graph of ( y = 3sinleft(2x + frac{pi}{4}right) ).Thus, the correct answer is:[boxed{text{C}}]

💡Alright, so I've got this problem about sequences and some terms called ( d_i ). It's divided into three parts, and I need to figure out each one step by step. Let me start with part (I).**Problem I: Given the sequence ( (a_n) ) as 3, 4, 7, 1, find the values of ( d_1 ), ( d_2 ), ( d_3 ).**Okay, so first, I need to understand what ( d_i ) represents. The problem says that for each ( i ) from 1 to ( n-1 ), ( A_i ) is the maximum of the first ( i ) terms, and ( B_i ) is the minimum of the subsequent terms from ( a_{i+1} ) to ( a_n ). Then, ( d_i = A_i - B_i ).Given the sequence is 3, 4, 7, 1. So, ( n = 4 ). Therefore, we need to find ( d_1 ), ( d_2 ), and ( d_3 ).Let me break it down:1. **Calculating ( d_1 ):** - ( i = 1 ) - First 1 term: [3] - Maximum ( A_1 = 3 ) - Subsequent terms: [4, 7, 1] - Minimum ( B_1 = 1 ) - So, ( d_1 = A_1 - B_1 = 3 - 1 = 2 )2. **Calculating ( d_2 ):** - ( i = 2 ) - First 2 terms: [3, 4] - Maximum ( A_2 = 4 ) - Subsequent terms: [7, 1] - Minimum ( B_2 = 1 ) - So, ( d_2 = A_2 - B_2 = 4 - 1 = 3 )3. **Calculating ( d_3 ):** - ( i = 3 ) - First 3 terms: [3, 4, 7] - Maximum ( A_3 = 7 ) - Subsequent terms: [1] - Minimum ( B_3 = 1 ) - So, ( d_3 = A_3 - B_3 = 7 - 1 = 6 )So, for part (I), the values are ( d_1 = 2 ), ( d_2 = 3 ), ( d_3 = 6 ). That seems straightforward.**Problem II: Given that ( (a_1) ), ( (a_2) ), ( ldots ), ( (a_{n-1}) ) (where ( n geq 4 )) is a geometric sequence with a common ratio greater than 1, and ( a_1 > 0 ). Prove that ( (d_1) ), ( (d_2) ), ( ldots ), ( (d_{n-1}) ) is a geometric sequence.**Alright, so now I need to prove that the sequence ( d_i ) is geometric given that ( a_i ) is geometric with a common ratio ( q > 1 ) and ( a_1 > 0 ).First, let's recall that in a geometric sequence, each term is the previous term multiplied by a common ratio ( q ). So, ( a_{k} = a_1 q^{k-1} ).Given that ( q > 1 ) and ( a_1 > 0 ), the sequence ( a_i ) is strictly increasing because each term is larger than the previous one.Now, let's think about ( A_i ) and ( B_i ):- ( A_i ) is the maximum of the first ( i ) terms. Since the sequence is increasing, the maximum will always be the last term, which is ( a_i ). - ( B_i ) is the minimum of the subsequent terms from ( a_{i+1} ) to ( a_n ). Since the sequence is increasing, the minimum of the subsequent terms will be the first term after ( a_i ), which is ( a_{i+1} ).Therefore, for each ( i ), ( A_i = a_i ) and ( B_i = a_{i+1} ). Hence, ( d_i = A_i - B_i = a_i - a_{i+1} ).But since ( a_{i+1} = a_i cdot q ), we can write:( d_i = a_i - a_i q = a_i (1 - q) ).Now, let's see if ( d_i ) forms a geometric sequence. Let's compute the ratio ( frac{d_{i+1}}{d_i} ):( frac{d_{i+1}}{d_i} = frac{a_{i+1}(1 - q)}{a_i(1 - q)} = frac{a_{i+1}}{a_i} = q ).Since the ratio between consecutive terms ( d_{i+1} ) and ( d_i ) is constant and equal to ( q ), the sequence ( d_i ) is indeed a geometric sequence with common ratio ( q ).Wait, but hold on. The problem says ( n geq 4 ), but does that affect anything? I don't think so because regardless of ( n ), as long as the sequence ( a_i ) is geometric with ( q > 1 ), the reasoning holds. So, I think this proof is solid.**Problem III: Given that ( (d_1) ), ( (d_2) ), ( ldots ), ( (d_{n-1}) ) is an arithmetic sequence with a common difference greater than 0, and ( d_1 > 0 ). Prove that ( (a_1) ), ( (a_2) ), ( ldots ), ( (a_{n-1}) ) is an arithmetic sequence.**Hmm, this seems a bit trickier. We need to show that if the ( d_i ) sequence is arithmetic, then the original ( a_i ) sequence is also arithmetic.Given:- ( d_i ) is arithmetic with common difference ( d > 0 ) and ( d_1 > 0 ).So, ( d_{i+1} = d_i + d ).We need to show that ( a_{i+1} - a_i ) is constant for all ( i ).Let me recall that ( d_i = A_i - B_i ). So, ( A_i = d_i + B_i ).But ( A_i ) is the maximum of the first ( i ) terms, and ( B_i ) is the minimum of the subsequent terms.Since ( d_i ) is increasing (because the common difference ( d > 0 )), ( A_i ) must be increasing as well because ( A_i = d_i + B_i ) and ( d_i ) is increasing.Wait, but ( B_i ) is the minimum of the subsequent terms. If ( A_i ) is increasing, does that tell us something about the ( a_i ) sequence?Let me think step by step.1. Since ( d_i ) is arithmetic, ( d_{i+1} = d_i + d ).2. ( A_{i+1} = d_{i+1} + B_{i+1} ).3. But ( A_{i+1} ) is the maximum of the first ( i+1 ) terms. Since ( A_i ) is the maximum of the first ( i ) terms, ( A_{i+1} = max(A_i, a_{i+1}) ).4. Similarly, ( B_{i} ) is the minimum of terms from ( a_{i+1} ) to ( a_n ), so ( B_{i+1} ) is the minimum of terms from ( a_{i+2} ) to ( a_n ).5. Since ( d_i ) is increasing, ( d_{i+1} = d_i + d ), so ( A_{i+1} = d_{i+1} + B_{i+1} = d_i + d + B_{i+1} ).6. Also, ( A_{i+1} = max(A_i, a_{i+1}) ).7. Let's assume that ( A_{i+1} = A_i ). Then, ( A_i = d_i + d + B_{i+1} ). But ( A_i = d_i + B_i ). So, ( d_i + B_i = d_i + d + B_{i+1} ), which simplifies to ( B_i = d + B_{i+1} ). But since ( B_{i+1} ) is the minimum of a subset of the terms that ( B_i ) is the minimum of, ( B_{i+1} geq B_i ). However, this would imply ( B_i = d + B_{i+1} geq B_i + d ), which is impossible because ( d > 0 ). Therefore, our assumption that ( A_{i+1} = A_i ) must be wrong.8. Hence, ( A_{i+1} = a_{i+1} ).So, ( A_{i+1} = a_{i+1} = d_{i+1} + B_{i+1} ).But ( A_i = d_i + B_i ).So, ( a_{i+1} = d_{i+1} + B_{i+1} ).But ( d_{i+1} = d_i + d ), so:( a_{i+1} = d_i + d + B_{i+1} ).But ( A_i = d_i + B_i ), so ( d_i = A_i - B_i ).Substituting back:( a_{i+1} = (A_i - B_i) + d + B_{i+1} ).But ( A_i ) is the maximum of the first ( i ) terms, which is ( a_i ) if the sequence is increasing. Wait, but we don't know yet if ( a_i ) is increasing. Hmm.Wait, let's think differently. Since ( d_i ) is increasing, and ( d_i = A_i - B_i ), and ( A_i ) is the maximum of the first ( i ) terms, which is non-decreasing. Similarly, ( B_i ) is the minimum of the remaining terms, which is non-increasing because as ( i ) increases, we're considering fewer terms, so the minimum can't decrease.But since ( d_i ) is increasing, the increase in ( A_i ) must be more significant than the decrease in ( B_i ).Wait, maybe it's better to express ( a_{i+1} ) in terms of ( a_i ).From earlier, we have:( a_{i+1} = d_{i+1} + B_{i+1} ).But ( d_{i+1} = d_i + d ), so:( a_{i+1} = d_i + d + B_{i+1} ).But ( d_i = A_i - B_i ), so:( a_{i+1} = (A_i - B_i) + d + B_{i+1} ).But ( A_i ) is the maximum of the first ( i ) terms, which is either ( A_{i-1} ) or ( a_i ). If the sequence is increasing, ( A_i = a_i ).Wait, but we don't know if the sequence is increasing yet. Hmm.Alternatively, let's consider that ( B_{i+1} geq B_i ) because it's the minimum of a smaller set. So, ( B_{i+1} geq B_i ).From the equation ( a_{i+1} = d_i + d + B_{i+1} ), since ( B_{i+1} geq B_i ), we have:( a_{i+1} geq d_i + d + B_i ).But ( d_i = A_i - B_i ), so:( a_{i+1} geq (A_i - B_i) + d + B_i = A_i + d ).But ( A_i ) is the maximum of the first ( i ) terms, which includes ( a_i ). So, ( A_i geq a_i ).Therefore, ( a_{i+1} geq A_i + d geq a_i + d ).This suggests that ( a_{i+1} geq a_i + d ), meaning the sequence ( a_i ) is increasing with a difference of at least ( d ).But we need to show that ( a_{i+1} - a_i ) is constant, which would make it an arithmetic sequence.Wait, let's see if we can express ( a_{i+1} ) in terms of ( a_i ).From earlier:( a_{i+1} = d_{i+1} + B_{i+1} ).But ( d_{i+1} = d_i + d ), so:( a_{i+1} = d_i + d + B_{i+1} ).But ( d_i = A_i - B_i ), so:( a_{i+1} = (A_i - B_i) + d + B_{i+1} ).Now, ( A_i ) is the maximum of the first ( i ) terms. If ( A_i = a_i ), then:( a_{i+1} = (a_i - B_i) + d + B_{i+1} ).But ( B_{i+1} ) is the minimum of the terms from ( a_{i+2} ) to ( a_n ). Hmm, this seems a bit tangled.Wait, maybe we can find a relationship between ( B_i ) and ( B_{i+1} ).Since ( B_i ) is the minimum of ( a_{i+1}, a_{i+2}, ldots, a_n ), and ( B_{i+1} ) is the minimum of ( a_{i+2}, ldots, a_n ), we have ( B_{i+1} geq B_i ).But from the equation ( a_{i+1} = (A_i - B_i) + d + B_{i+1} ), and since ( A_i geq a_i ), we can write:( a_{i+1} = (A_i - B_i) + d + B_{i+1} geq (a_i - B_i) + d + B_{i+1} ).But ( B_{i+1} geq B_i ), so:( a_{i+1} geq (a_i - B_i) + d + B_i = a_i + d ).This shows that ( a_{i+1} geq a_i + d ), meaning the sequence ( a_i ) is increasing with a common difference of at least ( d ).But we need to show that ( a_{i+1} - a_i ) is exactly ( d ), not just at least ( d ). How can we do that?Wait, let's consider the case when ( i = 1 ):( d_1 = A_1 - B_1 = a_1 - B_1 ).Since ( d_1 > 0 ), ( a_1 > B_1 ).Similarly, ( d_2 = d_1 + d = A_2 - B_2 ).But ( A_2 ) is the maximum of ( a_1, a_2 ). If ( a_2 > a_1 ), then ( A_2 = a_2 ). Otherwise, ( A_2 = a_1 ).But since ( d_2 = d_1 + d ), and ( d_1 = a_1 - B_1 ), we have:( d_2 = a_1 - B_1 + d ).But ( d_2 = A_2 - B_2 ).If ( A_2 = a_2 ), then:( a_2 - B_2 = a_1 - B_1 + d ).But ( B_2 ) is the minimum of ( a_3, ldots, a_n ), which is ( geq B_1 ) because ( B_1 ) is the minimum of ( a_2, ldots, a_n ).Wait, this is getting complicated. Maybe I should try to express ( a_{i+1} ) in terms of ( a_i ) and see if the difference is constant.From earlier, we have:( a_{i+1} = d_{i+1} + B_{i+1} ).But ( d_{i+1} = d_i + d ), so:( a_{i+1} = d_i + d + B_{i+1} ).But ( d_i = A_i - B_i ), so:( a_{i+1} = (A_i - B_i) + d + B_{i+1} ).Now, if ( A_i = a_i ) (which would be the case if the sequence is increasing), then:( a_{i+1} = (a_i - B_i) + d + B_{i+1} ).But ( B_{i+1} geq B_i ), so:( a_{i+1} geq a_i - B_i + d + B_i = a_i + d ).This shows that ( a_{i+1} geq a_i + d ).But we also have ( d_{i+1} = d_i + d ), so ( A_{i+1} - B_{i+1} = A_i - B_i + d ).If ( A_{i+1} = a_{i+1} ), then:( a_{i+1} - B_{i+1} = A_i - B_i + d ).But ( A_i = a_i ) (assuming the sequence is increasing), so:( a_{i+1} - B_{i+1} = a_i - B_i + d ).Rearranging:( a_{i+1} = a_i - B_i + d + B_{i+1} ).But ( B_{i+1} geq B_i ), so:( a_{i+1} geq a_i - B_i + d + B_i = a_i + d ).This again shows ( a_{i+1} geq a_i + d ).But we need to show equality, i.e., ( a_{i+1} = a_i + d ).Suppose, for contradiction, that ( a_{i+1} > a_i + d ). Then, ( A_{i+1} = a_{i+1} ), and:( d_{i+1} = A_{i+1} - B_{i+1} = a_{i+1} - B_{i+1} ).But ( d_{i+1} = d_i + d = (A_i - B_i) + d = (a_i - B_i) + d ).So,( a_{i+1} - B_{i+1} = a_i - B_i + d ).But if ( a_{i+1} > a_i + d ), then:( a_{i+1} - B_{i+1} > a_i + d - B_{i+1} ).But ( B_{i+1} geq B_i ), so:( a_i + d - B_{i+1} leq a_i + d - B_i ).Thus,( a_{i+1} - B_{i+1} > a_i + d - B_{i+1} geq a_i + d - B_i ).But ( d_{i+1} = a_i - B_i + d ), so:( a_{i+1} - B_{i+1} > d_{i+1} ).This contradicts the fact that ( d_{i+1} = a_{i+1} - B_{i+1} ).Therefore, our assumption that ( a_{i+1} > a_i + d ) must be false. Hence, ( a_{i+1} = a_i + d ).Thus, the sequence ( a_i ) is arithmetic with common difference ( d ).So, putting it all together, since ( a_{i+1} - a_i = d ) for all ( i ), the sequence ( a_1, a_2, ldots, a_{n-1} ) is arithmetic.**Summary:**- For part (I), calculating each ( d_i ) directly gives us the values.- For part (II), recognizing that ( a_i ) is geometric allows us to express ( d_i ) in terms of ( a_i ) and show the ratio is constant.- For part (III), using the properties of ( d_i ) being arithmetic and the relationships between ( A_i ) and ( B_i ) leads us to conclude that ( a_i ) must be arithmetic.I think that covers all parts of the problem.

💡Okay, so I have this equation here: (x - 5)^2 = 9. I need to find the sum of the roots. Hmm, let's see. First, I remember that when you have an equation like this, it's a quadratic equation because of the squared term. Quadratic equations usually have two solutions, or roots, and I need to find both of them and then add them together.Alright, so starting with the equation: (x - 5)^2 = 9. I think the first step is to solve for x. To do that, I can take the square root of both sides. Taking the square root of the left side, (x - 5)^2, would just give me (x - 5), right? And the square root of 9 is 3. But wait, I remember that when you take the square root of both sides, you have to consider both the positive and negative roots. So it should be x - 5 = ±3.Okay, so now I have two separate equations to solve:1. x - 5 = 32. x - 5 = -3Let's solve the first one: x - 5 = 3. To isolate x, I'll add 5 to both sides. So, x = 3 + 5, which means x = 8.Now, the second equation: x - 5 = -3. Again, I'll add 5 to both sides to solve for x. So, x = -3 + 5, which gives me x = 2.Alright, so the two roots are x = 8 and x = 2. Now, I need to find the sum of these roots. That means I just add them together: 8 + 2. Let's see, 8 plus 2 is 10. So, the sum of the roots is 10.Wait a second, let me make sure I didn't make a mistake. Maybe I should double-check my steps. Starting from the original equation: (x - 5)^2 = 9. Taking the square root of both sides gives x - 5 = ±3, which leads to x = 8 and x = 2. Adding those gives 10. Yeah, that seems right.Alternatively, I remember something called Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots. Maybe I can use that as a check. Let's see, if I expand the original equation, I can write it as a standard quadratic equation.Starting with (x - 5)^2 = 9, expanding the left side gives x^2 - 10x + 25 = 9. Then, subtracting 9 from both sides to set the equation to zero: x^2 - 10x + 16 = 0. So, the quadratic equation is x^2 - 10x + 16 = 0.According to Vieta's formulas, the sum of the roots of a quadratic equation ax^2 + bx + c = 0 is -b/a. In this case, a is 1 and b is -10. So, the sum of the roots should be -(-10)/1, which is 10. That matches what I got earlier by solving directly. So, that confirms that the sum of the roots is indeed 10.I think I'm confident now that the sum of the roots is 10. It makes sense both by directly solving the equation and by using Vieta's formulas. I don't see any mistakes in my reasoning, so I'll stick with that answer.

💡First, I need to determine how many team members showed up to the game. Each member who attended scored 5 points, and the team's total score was 35 points.To find the number of members who showed up, I'll divide the total points by the points scored per member:35 points ÷ 5 points/member = 7 members showed up.Next, I know that 7 members did not show up to the game. To find the total number of team members, I'll add the number of members who showed up to the number who did not show up:7 members (showed up) + 7 members (did not show up) = 14 members in total.Therefore, the trivia team has 14 members in total.