Question

Find the value of ‘$p$’ for which the roots of the following equations are real and equal:

$4x^2+px+9=0$

Answer 1

Step 1:

Comparison of the coefficients:

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

$\begin{array}{rcl}a& =& 4\\ b& =& p\\ c& =& 9\end{array}$

Step 2:

Find the discriminant:

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

$\begin{array}{rcl}D& =& b^2-4ac\\ & =& {\left(p\right)}^2-\left(4\times 4\times 9\right)\\ & =& p^2-144\end{array}$

Therefore, the discriminant is $D=p^2-144$.

For real and equal roots, the discriminant must be equal to 0 $\left(D=0\right)$.

$$\begin{gathered} \begin{array}{rcl}{p}^2-144& =& 0\\ & \Rightarrow & p^2=144\\ & \Rightarrow & p=\pm \sqrt{144}\\ & \Rightarrow & p=\pm 12\end{array} \\ \Rightarrow P=12,-12 \end{gathered}$$

Hence, the value of $p$ is $12$ and $-12$.