Question

If $x-5$ is the factor of quadratic equation $2x^2+px-13=0$ and the quadratic equation $p(x^2+x)+k=0$ . Find the value of k.

Answer 1

$2x^2+px-13=0$ ---- ( 1 )

$p(x^2+x)+k=0$ ---- ( 2 )

$x-5$ is roots of the equation.

$\therefore$ substituting $x=5$ in given quadratic equation ( 1 )

$\Rightarrow$ $2(5{)}^2+p(5)-13=0$

$\Rightarrow$ $50+5p-13=0$

$\Rightarrow$ $37+5p=0$

$\therefore$ $p=\frac{-37}{5}$

Now, substituting $x=5$ and $p=\frac{-37}{5}$ in equation ( 2 )

$\Rightarrow$ $\frac{-37}{5}\left[\right(5{)}^2+5\left)\right]+k=0$

$\Rightarrow$ $\frac{-37}{5}[25+5]+k=0$

$\Rightarrow$ $-37\left(30\right)+5k=0$

$\Rightarrow$ $-1110+5k=0$

$\Rightarrow$ $5k=1110$

$\therefore$ $k=222$