Question

Find the value of $p$ for which the equation: $p{x}^2-8x+4=0$ has two equal roots.

Answer 1

Step 1:

Comparison of the coefficients:

Compare the coefficients of given equation with standard equation, $a{x}^2+bx+c=0$.

$\begin{array}{rcl}a& =& p\\ b& =& -8\\ c& =& 4\end{array}$

Step 2:

Find the discriminant:

The discriminant $D$ of the given equation can be calculated as,

$\begin{array}{rcl}D& =& b^2-4ac\\ & =& {\left(-8\right)}^2-\left(4\times p\times 4\right)\\ & =& 64-16p\end{array}$

Since, the given equation has equal roots, so the discriminant is zero.

$$\begin{gathered} D=0 \\ \begin{array}{rcl}& \Rightarrow & 64-16p=0\\ & \Rightarrow & 16p=64\\ & \Rightarrow & p=\frac{64}{16}\\ & \Rightarrow & p=4\end{array} \end{gathered}$$

Hence, the required value of $p=4$.