Question

Find k, if the sum of the roots of the quadratic equation 4x2 + 8kx + k + 9 = 0 is equal to their product.

Answer 1

Given: 4x² + 8kx + k + 9 = 0
On comparing this equation with ax² + bx + c = 0, we get:
a = 4, b = 8k and c = k + 9
Let α and β be the roots of the given quadratic equation. Then, we have
$\alpha +\beta =\alpha \beta$ …(1)
We know that α + β = –$\frac{b}{a}$ and αβ = $\frac{c}{a}$.
Thus,
$$\begin{gathered} \frac{-b}{a}=\frac{c}{a} \\ \Rightarrow -b=c \\ \Rightarrow b=-c \end{gathered}$$
On substituting b = 8k and c = k + 9, we get:
8k = –(k + 9)
$\Rightarrow$ 8k = –k – 9
$\Rightarrow$8k + k = –9
$\Rightarrow$9k = –9
$\Rightarrow$k = –1