Question

Form the quadratic equation from the roots given below.
(1) 0 and 4
(2) 3 and –10
(3) $\frac{1}{2},-\frac{1}{2}$
(4) $2-\sqrt{5},2+\sqrt{5}$

Answer 1

(1) 0 and 4
Sum of roots = 0 + 4 = 4
Product of roots = 0 $\times$ 4 = 0
The general form of the quadratic equation is $x^2-\left(\mathrm{Sum}\mathrm{of}\mathrm{roots}\right)x+\mathrm{Product}\mathrm{of}\mathrm{roots}=0$
So, the quadratic equation obtained is $x^2-4x+0=0$
$$\begin{gathered} \Rightarrow x^2-4x=0 \\ \Rightarrow x\left(x-4\right)=0 \end{gathered}$$

(2) 3 and –10
Sum of roots = 3 + (–10) = −7
Product of roots = 3 $\times$ –10 = –30
The general form of the quadratic equation is $x^2-\left(\mathrm{Sum}\mathrm{of}\mathrm{roots}\right)x+\mathrm{Product}\mathrm{of}\mathrm{roots}=0$
So, the quadratic equation obtained is $x^2-\left(-7\right)x+\left(-30\right)=0$
$x^2+7x-30=0$

(3) $\frac{1}{2},-\frac{1}{2}$
Sum of roots = $\frac{1}{2}+\left(-\frac{1}{2}\right)=0$
Product of roots = $\frac{1}{2}\times \left(\frac{-1}{2}\right)=\frac{-1}{4}$
The general form of the quadratic equation is $x^2-\left(\mathrm{Sum}\mathrm{of}\mathrm{roots}\right)x+\mathrm{Product}\mathrm{of}\mathrm{roots}=0$
So, the quadratic equation obtained is $x^2-\left(0\right)x+\left(\frac{-1}{4}\right)=0$
$\Rightarrow x^2-\frac{1}{4}=0$

(4) $2-\sqrt{5},2+\sqrt{5}$
Sum of roots = $2-\sqrt{5}+2+\sqrt{5}=4$
Product of roots = $\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)=4-5=-1$
The general form of the quadratic equation is $x^2-\left(\mathrm{Sum}\mathrm{of}\mathrm{roots}\right)x+\mathrm{Product}\mathrm{of}\mathrm{roots}=0$
So, the quadratic equation obtained is $x^2-\left(4\right)x+\left(-1\right)=0$
$\Rightarrow x^2-4x-1=0$