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

The Correct Option is C

\(x^2 – (A – 3)x – (A – 7) = 0 \)
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.