Question

For each equation given below, find the value (s) of $p$ so that the equation has equal roots:

$p{x}^2+4x+p=0$

Answer 1

Step 1: Solve for discriminant.

The given equation is as follows:

$p{x}^2+4x+p=0$

Compare the given quadratic equation with $a{x}^2+bx+c=0$ to get:

$a=p,b=4,c=p$

Now, discriminant can be calculated as:

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

Step 2: Apply the condition for equal roots.

For equal roots,

$\begin{array}{rcl}{b}^2-4ac& =& 0\\ 16-4p^2& =& 0\\ 16& =& 4p^2\\ p^2& =& \frac{16}{4}\\ p& =& \sqrt{4}\\ p& =& \pm 2\end{array}$

Thus for $p=\pm 2$, the equation has equal roots.