Question

For what value of value of $k,$ does the quadratic equation $9x^2+8kx+16=0$ have equal roots.

Answer 1

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

Given that, equation $9x^2+8kx+16=0$ has equal roots.

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

$D=B^2-4AC$

For equation $9x^2+8kx+16=0$ the value $A=9,B=8k$ and $C=16$.

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

$\begin{array}{rcl}D& =& B^2-4AC\\ & =& {\left(8k\right)}^2-4\left(9\right)\left(16\right)\\ & =& 64\left[k^2-9\right]\end{array}$

Step 2: Determining the values of $k$

According to the nature of the 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,

$\begin{array}{rcl}D& =& 0\\ & \Rightarrow & 64\left[k^2-9\right]=0\\ & \Rightarrow & k^2-9=0\\ & \Rightarrow & k=\pm 3\end{array}$

So,

$k=-3$ or $k=3$

Hence, for $k=-3,3$ the equation $9x^2+8kx+16=0$ has equal roots.