Question

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients :

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

Answer 1

$2\sqrt{3}{x}^2-5x+\sqrt{3}=0$

$2\sqrt{3}{x}^2-(3+2)x+\sqrt{3}=0$

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

$\sqrt{3}x(2x-\sqrt{3})-1(2x-\sqrt{3})=0$

$(2x-\sqrt{3})(\sqrt{3}x-1)=0$

then

$2x-\sqrt{3}=0$

$x=\frac{\sqrt{3}}{2}$

and $\sqrt{3}x-1=0$

$x=\frac{1}{\sqrt{3}}$

Zeroes are $\frac{\sqrt{3}}{2}and\frac{1}{\sqrt{3}}$

Sum of zeros $\frac{-b}{a}=\frac{-(-5)}{2\sqrt{3}}=\frac{5}{2\sqrt{3}}$

Sum of zeros $\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{3}}=\frac{3+2}{2\sqrt{3}}=\frac{5}{2\sqrt{3}}$

Product of zeros $\frac{c}{a}=\frac{\sqrt{3}}{2\sqrt{3}}=\frac{1}{2}$

Product of zeros $\frac{\sqrt{3}}{2}\times \frac{1}{\sqrt{3}}=\frac{1}{2}$

hence proved that

Sum of zeros $\frac{-\left(Coefficientofx\right)}{Coefficientof{x}^2}$

Product of zeros $\frac{Constantterm}{Coefficientof{x}^2}$