Question

Obtain a quadratic polynomial $p\left(x\right)=a{x}^2+bx+c$, where sum of zeros is $\sqrt{2}$ and product of zeros is $\frac{1}{3}$; $a>0$

Answer 1

$p\left(x\right)=a{x}^2+bx+c$-Equation 1

Given sum of roots$=\alpha +\beta =\frac{-b}{a}=\sqrt{2}$-Equation 2

and Product of roots$=\alpha \cdot \beta =\frac{c}{a}=\frac{1}{3}$(given $a>0$)-Equation 3

For roots, let's put $p\left(x\right)=0$

$\Rightarrow a{x}^2+bx+c=0$

$\Rightarrow x^2+\frac{b}{a}x+\frac{c}{a}=0$

By putting values from Equation 2 & 3, we get

$\Rightarrow x^2-\sqrt{2}x+\frac{1}{3}=0$

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