Question

Find the zeroes of the following polynomial by factorisation method and verify the relation between the zeroes and the coefficients of the polynomials:

$y^2+\frac{3}{2}\sqrt{5}y-5$.

• A. $-2\sqrt{5},\frac{\sqrt{5}}{2}$
• B. $-2\sqrt{5},\frac{\sqrt{3}}{2}$
• C. $-2\sqrt{5},\frac{\sqrt{11}}{2}$
• D. $-2\sqrt{5},\frac{\sqrt{10}}{2}$

Answer 1

The correct option is A $-2\sqrt{5},\frac{\sqrt{5}}{2}$

The quadratic polynomial is $y^2+\frac{3}{2}\sqrt{5}y-5$.
The zeros of the polynomial are:
$y^2+\frac{3}{2}\sqrt{5}y-5=0$
$⟹$ $y^2+2\sqrt{5}y-\frac{\sqrt{5}}{2}y-5=0$
$⟹$ $y(y+2\sqrt{5})-\frac{\sqrt{5}}{2}(y+2\sqrt{5})=0$
$⟹$ $(y+2\sqrt{5})(y-\frac{\sqrt{5}}{2})=0$
$⟹$ $y-2\sqrt{5}$ and $y=\frac{\sqrt{5}}{2}=0$
The zeros are $-2\sqrt{5}$ and $\frac{\sqrt{5}}{2}$.

Now verifying the relation of zeros with the coefficients of the quadratic polynomial $y^2+\frac{3}{2}\sqrt{5}y-5$:

Comparing it with the standard from of quadratic equation $a{x}^2+bx+c$, we get, $a=1$, $b=\frac{3}{2}\sqrt{5}$ and $c=-5$
Thus,
Sum of the zeros $=-2\sqrt{5}+\frac{\sqrt{5}}{2}$ $=\frac{-4\sqrt{5}+\sqrt{5}}{2}$ $=\frac{-3\sqrt{5}}{2}$
$=-\frac{b}{a}$.
Product of the zeros $=-2\sqrt{5}\times \frac{\sqrt{5}}{2}$ $=-5$
$=\frac{c}{a}$.

Hence, verified.

Therefore, option $A$ is correct.