Question

Given the quadratic equation $x^2 – (A – 3)x – (A – 7) = 0$, for what value of A is the sum of squares of the roots 0?

• A. -2
• B. 3
• C. 5
• D. 10
• E. 13

Answer 1

Answer: (C)

5

Answer 2

$x^2 – (A – 3)x – (A – 7) = 0$
The sum of the roots $(α + β )$ and the product of the roots $(α β )$for a quadratic equation, $ax^2 + bx + c = 0$, is given by
$α + β = -\frac{b}{a} = A – 3$
$α β = \frac{c}{a} = – (A – 7)$
According to the question,
$α^2 \+ β^2 = 0$
$(α + β )^2 – 2α β = 0$
$(A – 3)^2 \+ 2(A – 7) = 0$
$A^2 – 6A + 9 + 2A – 14 = 0$
$A^2 – 4A – 5 = 0$
$A^2 – 5A + A – 5 = 0$
$A(A – 5) + 1(A – 5) = 0$
$(A – 5)(A + 1) = 0$
$A = 5\space or \space–1$
$\therefore$ The correct answer is C.