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Given the quadratic equation $x^{2}-(A-3) x- (A-2) = 0$, for what value of $A$ will the sum of the squares of the roots be zero?

1. $-2$
2. $3$
3. $6$
4. None of these
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+1 vote

Let the two roots be $\alpha _{1}$ and $\alpha _{2}$

For a quadratic equation $ax^2 +bx +c = 0$, sum of the roots $= -\frac{b}{a}$ and product of the roots $=\frac{c}{a}.$So,

$\alpha _{1}$ + $\alpha _{2}$ = (A - 3)

$\alpha _{1}$.$\alpha _{2}$ = -(A - 2)

We want $\alpha _{1}^{2}$ + $\alpha _{2}^{2}$ = 0

$\Rightarrow$ $\left ( \alpha _{1} + \alpha _{2} \right )^{2}$ - 2$\alpha _{1}$.$\alpha _{2}$ = 0

$\Rightarrow$ (A - 3)2 - 2(-(A - 2)) = 0

$\Rightarrow$ A2 - 4A + 5 = 0

$\Rightarrow$ A = 2 $\pm$ i

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