# Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

Question:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients

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

Solution:

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

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

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

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

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

$\Rightarrow x=\frac{1}{\sqrt{3}}$ or $x=\frac{\sqrt{3}}{2}$

$\Rightarrow x=\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}$ or $x=\frac{\sqrt{3}}{2}$

Sum of zeroes $=\frac{\sqrt{3}}{3}+\frac{\sqrt{3}}{2}=\frac{5 \sqrt{3}}{6}=\frac{-(\text { coefficient of } x)}{\text { coefficient of } x^{2}}$

Product of zeroes $=\frac{\sqrt{3}}{3} \times \frac{\sqrt{3}}{2}=\frac{1}{2}=\frac{\text { constant term }}{\text { coefficient of } x^{2}}$