Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients
$2 \sqrt{3} x^{2}-5 x+\sqrt{3}$
$2 \sqrt{3} x^{2}-5 x+\sqrt{3}$
$\Rightarrow 2 \sqrt{3} x^{2}-2 x-3 x+\sqrt{3}$
$\Rightarrow 2 x(\sqrt{3} x-1)-\sqrt{3}(\sqrt{3} x-1)=0$
$\Rightarrow(\sqrt{3} x-1)$ 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 $=\frac{\sqrt{3}}{3} \times \frac{\sqrt{3}}{2}=\frac{1}{2}=\frac{\text { constant term }}{\text { coefficient of } x^{2}}$
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