Question

Given the quadratic equation \( x^2 - (A - 3)x - (A - 2) \), for what value of \( A \) will the sum of the squares of the roots be zero?

• A. -2
• B. 3
• C. 6
• D. None of these

Hint:

In problems involving quadratic equations, use Vieta's formulas to relate the coefficients to the roots, and apply the sum of squares formula to derive the answer.

Answer 1

The Correct Option is D

The given quadratic equation is: \[ x^2 - (A - 3)x - (A - 2) = 0 \] Let the roots of the quadratic equation be \( r_1 \) and \( r_2 \). Using Vieta's formulas, we know the following relationships between the coefficients and the roots: - The sum of the roots is \( r_1 + r_2 = A - 3 \), - The product of the roots is \( r_1 r_2 = -(A - 2) \). We are asked to find the value of \( A \) such that the sum of the squares of the roots is zero. The sum of the squares of the roots is given by: \[ r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 \] Substituting the known values from Vieta's formulas: \[ r_1^2 + r_2^2 = (A - 3)^2 - 2(-(A - 2)) = (A - 3)^2 + 2(A - 2) \] Simplifying the expression: \[ r_1^2 + r_2^2 = (A^2 - 6A + 9) + 2A - 4 = A^2 - 4A + 5 \] We are given that the sum of the squares of the roots is zero, so: \[ A^2 - 4A + 5 = 0 \] Solving this quadratic equation using the quadratic formula: \[ A = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(5)}}{2(1)} = \frac{4 \pm \sqrt{16 - 20}}{2} = \frac{4 \pm \sqrt{-4}}{2} \] Since the discriminant is negative, there is no real solution for \( A \). Thus, the sum of the squares of the roots cannot be zero for any real value of \( A \). Therefore, the answer is d. None of these.