Question

Sum of the roots of a quadratic equation is double their product. Find k if equation x² – 4kx + k +3 = 0

Answer 1

Let the roots of the quadratic equation be $\alpha \mathrm{and}\beta$.
Sum of roots = $\alpha +\beta$
Product of roots = $\alpha \beta$
Given that sum of roots = double the product
$\Rightarrow \alpha +\beta =2\alpha \beta$ .....(I)
Given: x² – 4kx + k +3 = 0 .....(II)
General form of the quadratic equation is $x^2-\left(\mathrm{Sum}\mathrm{of}\mathrm{roots}\right)x+\mathrm{Product}\mathrm{of}\mathrm{roots}=0$
$$\begin{gathered} \Rightarrow x^2-\left(\alpha +\beta \right)x+\alpha \beta =0 \\ \Rightarrow x^2-\left(2\alpha \beta \right)x+\alpha \beta =0.....\left(\mathrm{III}\right) \end{gathered}$$
On comparing (II) and (III) we have
$4k=2\alpha \beta \mathrm{and}k+3=\alpha \beta$
Solving these equations we have
$$\begin{gathered} 4k=2\left(k+3\right) \\ \Rightarrow 2k=k+3 \\ \Rightarrow 2k-k=3 \\ \Rightarrow k=3 \end{gathered}$$