Question

Find the value of '$p$' for which the quadratic equation has equal roots : $2a^2+3a+p=0$

Answer 1

We know that while finding the root of a quadratic equation $a{x}^2+bx+c=0$ by quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$,

if $b^2-4ac>0$, then the roots are real and distinct

if $b^2-4ac=0$, then the roots are real and equal and

if $b^2-4ac<0$, then the roots are imaginary.

Here, the given quadratic equation $2a^2+3a+p=0$ is in the form $a{x}^2+bx+c=0$ where $a=2,b=3$ and $c=p$.

It is given that the roots are equal, therefore $b^2-4ac=0$ that is:

$b^2-4ac=0\Rightarrow (3{)}^2-(4\times 2\times p)=0\Rightarrow 9-8p=0\Rightarrow -8p=-9\Rightarrow 8p=9\Rightarrow p=\frac{9}{8}$

Hence, $p=\frac{9}{8}$