Multiple choice

Directions: Solve the following question and mark the best possible option. The minimum possible value of the sum of the squares of the roots of the equation x2 + (a + 3)x - (a + 5) = 0 is

  1. 1

  2. 2

  3. 3

  4. 4
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C Correct answer
Explanation

Sum of roots alpha+beta = -(a+3). Product of roots alpha*beta = -(a+5). Sum of squares = (alpha+beta)^2 - 2*alpha*beta = (a+3)^2 + 2(a+5) = a^2 + 6a + 9 + 2a + 10 = a^2 + 8a + 19. To minimize, take derivative: 2a + 8 = 0 => a = -4. Minimum value = (-4)^2 + 8(-4) + 19 = 16 - 32 + 19 = 3.

AI explanation

Let the roots of the equation be alpha and beta. Using the sum and product of roots formulas, we get alpha + beta = -(a + 3) and alpha * beta = -(a + 5). We need to minimize the expression alpha squared plus beta squared, which can be rewritten using the algebraic identity as the quantity of alpha plus beta squared minus two times alpha times beta. Substituting the values gives the quantity of negative (a + 3) squared minus two times negative (a + 5), which simplifies to a squared + 6a + 9 + 2a + 10, and further to a squared + 8a + 19. To find the minimum value of this quadratic expression, we calculate the vertex using the formula negative b divided by 2a, giving a = -8 divided by 2, which is -4. Substituting a = -4 into the expression yields 16 - 32 + 19, resulting in 3.

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