Question

Find a quadratic polynomial whose sum and product respectively of the zeros, are $\frac{-3}{2\sqrt{5}},-\frac{1}{2}$.

• A. Polynomial: $x^2+3\sqrt{5}x-7\sqrt{5}$
• B. Polynomial: $2\sqrt{5}{x}^2+3x-\sqrt{5}$
• C. Polynomial: $2x^2-7\sqrt{5}x-3\sqrt{5}$
  
• D. None of these

Answer 1

Answer: (B)

Polynomial: $2\sqrt{5}{x}^2+3x-\sqrt{5}$

Given,

$\Rightarrow$Sum of zeros : $S=\frac{-3}{2\sqrt{5}}$

and

$\Rightarrow$Product of zeros : $P=-\frac{1}{2}$

$\Rightarrow$The required quadratic equation $p\left(x\right)=(x^2-Sx+P)$, where $k$ is real constant.

$\Rightarrow p\left(x\right)=[x^2+\frac{3}{2\sqrt{5}}x-\frac{1}{2}]$

Now,$P\left(x\right)=0$

$\therefore 0=[x^2+\frac{3}{2\sqrt{5}}x-\frac{1}{2}]$

Multiply by $2\sqrt{5}$

$\Rightarrow$$Polynomial:2\sqrt{5}{x}^2+3x-\sqrt{5}$