Question

A quadratic equation whose difference of roots is $3$ and the sum of the squares of the roots is $29,$ is given by

• A. $x^2+9x+14=0$
• B. $x^2+7x+10=0$
• C. $x^2-7x-10=0$
• D. $x^2-7x+10=0$

Answer 1

Answer: (B), (D)

The correct options are
B $x^2+7x+10=0$
D $x^2-7x+10=0$

Let $\alpha$ and $\beta$ be the roots of required quadratic equation. Then
${\alpha }^2+{\beta }^2=29$
and $|\alpha -\beta |=3$
Squaring both sides,
${\alpha }^2+{\beta }^2-2\alpha \beta =9\Rightarrow \alpha \beta =\frac{29-9}{2}\Rightarrow \alpha \beta =10$

Then, $(\alpha +\beta {)}^2={\alpha }^2+{\beta }^2+2\alpha \beta =29+20$
$\Rightarrow \alpha +\beta =\pm 7$

$\therefore$ Required equation is
$x^2-(\alpha +\beta )x+\alpha \beta =0$
$\Rightarrow x^2\pm 7x+10=0$