Question

Find the value of $k$ for which the equation $9x^2+3kx+4=0$ have equal roots.

Answer 1

Step 1:

Comparison of the coefficients:

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

$$\begin{gathered} a=9 \\ b=3k \\ c=4 \end{gathered}$$

Step 2:

Find the discriminant:

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

$\begin{array}{rcl}D& =& b^2-4ac\\ & =& {\left(3k\right)}^2-\left(4\times 9\times 4\right)\\ & =& 9k^2-144\end{array}$

Since, the given equation has equal roots, so the discriminant must be equal to zero.

$\begin{array}{rcl}D& =& 0\\ 9k^2-144& =& 0\\ 9k^2& =& 144\\ k^2& =& \frac{144}{9}\\ k& =& \sqrt{\frac{144}{9}}\\ k& =& \pm 4\end{array}$

Hence, the required value is $k=4\mathrm{and}k=-4$.