# Find a quadratic polynomial whose sum and product respectively of the zeroes are as given .Also find the zeroes of these polynomials by factorization ((-3)/(2sqrt5)), (-1/2)

Find a quadratic polynomial whose sum and product respectively of the zeroes are as given .Also find the zeroes of these polynomials by factorization
$\left(\frac{-3}{2\sqrt{5}}\right),\left(-\frac{1}{2}\right)$
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Step 1
Given:
$\left(\frac{-3}{2\sqrt{5}}\right),\left(-\frac{1}{2}\right)$
Sum of the zeroes $=-\frac{3}{2}\sqrt{5}x$
Product of the zeroes $=-\frac{1}{2}$
Step 2 Finding the zeroes
$p\left(x\right)={x}^{2}$ -(sum of the zeroes)+(product of the zeroes)
$p\left(x\right)={x}^{2}-\frac{3}{2}\sqrt{5}x-\frac{1}{2}$
$p\left(x\right)=2\sqrt{5}{x}^{2}-3x-\sqrt{5}$
$2\sqrt{5}{x}^{2}-3x-\sqrt{5}=0$
$2\sqrt{5}{x}^{2}-\left(5x-2x\right)-\sqrt{5}=0$
$2\sqrt{5}{x}^{2}-5x+2x-\sqrt{5}=0$
$\sqrt{5}x\left(2x-\sqrt{5}\right)-\left(2x-\sqrt{5}\right)=0$
$\left(2x-\sqrt{5}\right)\left(\sqrt{5}x-1\right)=0$
$2x-\sqrt{5}=0,\sqrt{5}x-1=0$
$2x=\sqrt{5},\sqrt{5}x=1$
$x=\frac{\sqrt{5}}{2},x=\frac{1}{\sqrt{5}}$
$\therefore x=\frac{\sqrt{5}}{2},\frac{1}{\sqrt{5}}$