Question

Form equations whose roots are
(i) $3,-\frac{1}{3}$ (ii) $\sqrt{3},\frac{1}{\sqrt{3}}$ (iii) $0,0$

Answer 1

We know that a quadratic Equation can be written as

$x^2-\left(SumofRoots\right)x+\left(ProductofRoots\right)=0$

Now,

(i) if roots are $3,-\frac{1}{3}$

So, the Equation having these roots will be

$=x^2-(3-\frac{1}{3})x+(-\frac{1}{3}).\left(3\right)=0\Rightarrow x^2-\frac{8}{3}x-1=0\Rightarrow 3x^2-8x-3=0$

(ii) if roots are $\sqrt{3},-\frac{1}{\sqrt{3}}$

So, the Equation having these roots will be

$=x^2-(\sqrt{3}+\frac{1}{\sqrt{3}})x+\left(\frac{1}{\sqrt{3}}\right).\left(\sqrt{3}\right)=0\Rightarrow x^2-\frac{4}{\sqrt{3}}x+1=0\Rightarrow \sqrt{3}{x}^2-4x+\sqrt{3}=0$

(iii) if roots are $0,0$

So, the Equation having these roots will be

$=x^2-(0+0)x+\left(0\right).\left(0\right)=0\Rightarrow x^2=0$