Question

If $-5$ is a root of the quadratic equation $2x^2+px-15=0$ and the quadratic equation $p(x^2+x)+k=0$ has equal roots, find the value of $k$.

Answer 1

Since, $-5$ is a root of the quadratic equation $2x^2+px-15=0$, then,

$\Rightarrow 2(-5{)}^2+p(-5)-15=0$

$\Rightarrow 50-5p-15=0$

$\Rightarrow 5p=35$

$\therefore p=7$

Hence, $p(x^2+x)+k=0$ becomes,

$\Rightarrow 7x^2+7x+k=0$

Since, this quadratic equation has equal roots, therefore,

$\Rightarrow D=b^2-4ac=0$

$\Rightarrow (7{)}^2-4(7\left)\right(k)=0$

$\Rightarrow 28k=49$

$\therefore k=\frac{7}{4}$