Question

Two roots of quadratic equations are given ; frame the equation.
(1) 10 and –10
(2) $1-3\sqrt{5}\mathrm{and}1+3\sqrt{5}$
(3) 0 and 7

Answer 1

(1) 10 and –10
Sum of roots = 10 + (–10) = 0
Product of roots = $10\times \left(-10\right)=-100$
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 will be
$$\begin{gathered} x^2-0x-100=0 \\ \Rightarrow x^2-100=0 \end{gathered}$$

(2) $1-3\sqrt{5}\mathrm{and}1+3\sqrt{5}$
Sum of roots = $1-3\sqrt{5}+1+3\sqrt{5}=2$
Product of roots = $\left(1-3\sqrt{5}\right)\left(1+3\sqrt{5}\right)=1-45=-44$
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 will be
$x^2-2x-44=0$

(3) 0 and 7
Sum of roots = 0 + 7 = 7
Product of roots = $0\times 7=0$
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 will be
$$\begin{gathered} x^2-7x+0=0 \\ \Rightarrow x^2-7x=0 \end{gathered}$$