Question

Find k, if one of the roots of the quadratic equation kx² – 7x + 12 = 0 is 3.

Answer 1

Given: kx² – 7x + 12 = 0
On comparing this equation with ax² + bx + c = 0, we get:
a = k, b = –7 and c = 12
Let α and β be the roots of the quadratic equation kx² – 7x + 12 = 0. Then, α = 3.
We know that α + β = – $\frac{b}{a}$ and αβ = $\frac{c}{a}$.
From α + β = – $\frac{b}{a}$, we get:
3 + β = $-\frac{-7}{k}=\frac{7}{k}$
$\Rightarrow$β = $\frac{7}{k}$ – 3 …(1)

From αβ = $\frac{c}{a}$, we get:
3 ×β = $\frac{12}{k}$
$\Rightarrow$ β = $\frac{4}{k}$
From equation (1), we get:
$$\begin{gathered} \frac{7}{k}-3=\frac{4}{k} \\ \Rightarrow \frac{7}{k}-\frac{4}{k}=3 \\ \Rightarrow \frac{7-4}{k}=3 \\ \Rightarrow \frac{3}{k}=3 \\ \Rightarrow k=1 \end{gathered}$$