Question

Find k, if the roots of the quadratic equation x² + kx + 40 = 0 are in the ratio 2:5

Answer 1

Given: x² + kx + 40 = 0
On comparing this equation with ax² + bx + c = 0, we get:
a = 1, b = k and c = 40
Let α and β be the roots of the quadratic equation x² + kx + 40 = 0 Then,
$$\begin{gathered} \alpha :\beta =2:5 \\ \Rightarrow \frac{\alpha }{\beta }=\frac{2}{5} \end{gathered}$$
Now, we let α = 2z and β = 5z, where z is a constant.
We know that α + β = – $\frac{b}{a}$ and αβ = $\frac{c}{a}$.
From α + β = – $\frac{b}{a}$, we get:
2z + 5z = $\frac{-k}{1}$ = $-$ k
$\Rightarrow$7z = $-$k
$\Rightarrow$ z = $\frac{-k}{7}$…(1)
From αβ = $\frac{c}{a}$, we get:
2z × 5z = = $\frac{40}{1}$
$\Rightarrow$ 10z² = 40
$\Rightarrow$z² = 4
$\Rightarrow$ z = –2 , 2

Now, on substituting z = –2 in equation (1), we get:
$$\begin{gathered} -2=\frac{-k}{7} \\ \Rightarrow k=14 \end{gathered}$$
And, on substituting z = 2 in equation (1), we get:
$2=\frac{-k}{7}\Rightarrow k=-14$
Therefore, k = ± 14.