Question

Find the value of $p$ for which the quadratic equation $2p{x}^2-8x+p=0$ has equal roots.

Answer 1

Step 1: Determining the discriminant $\left(D\right)$ of the quadratic equation

Given that, equation $2p{x}^2-8x+p=0$ has equal roots.

The discriminant of the quadratic equation is determined by using the formula:

$D=B^2-4AC$

For equation $2p{x}^2-8x+p=0$ the value $A=2p,B=-8$ and $C=p$.

Substituting the values of $A$, $B$ and $C$ in discriminant formula.

$\begin{array}{rcl}D& =& B^2-4AC\\ & =& {\left(-8\right)}^2-4\left(2p\right)\left(p\right)\\ & =& 64-8p^2\\ & =& 8(8-p^2)\end{array}$

Step 2: Determine the values of $p$

According to the nature of roots of a quadratic equation if the discriminant $D=0$, then the quadratic equation has real and equal roots.

By nature of roots of the equation,

$\begin{array}{rcl}D& =& 0\\ & \Rightarrow & 8\left(8-p^2\right)=0\\ & \Rightarrow & p^2=8\\ & \Rightarrow & p=\pm 2\sqrt{2}\end{array}$

Hence, for $p=-2\sqrt{2},2\sqrt{2}$ the equation $2p{x}^2-8x+p=0$ has equal roots.