Question

Find the value of $k$ such that the sum of the squares of the roots of the quadratic equation $x^2-8x+k=0$ is $40$.

• A. $12$
• B. $2$
• C. $5$
• D. $8$

Answer 1

Answer: (A)

$12$

The given equation is, $x^2-8x+k=0$
Here $a=1,b=-8,c=k$
Let $\alpha ,\beta$ be the roots of the given equation.

Then, sum of the roots $(\alpha +\beta )=-\frac{b}{a}$ $=8$ and
product of the roots $\left(\alpha \beta \right)=\frac{c}{a}$ $=k$
Now, sum of the squares of the roots $=$ $40$.
$\Rightarrow {\alpha }^2+{\beta }^2=40$
$\Rightarrow {\alpha }^2+{\beta }^2+2\alpha \beta -2\alpha \beta =40$
$\Rightarrow (\alpha +\beta {)}^2-2\alpha \beta =40$
$\Rightarrow 8^2-2\times k=40$
$\Rightarrow 64-2k=40$
$\Rightarrow -2k=40-64$
$\Rightarrow -2k=-24$
$\Rightarrow k=12$