Question

Find the zeroes of the following quadratic polynomial and verify.
$x^2-2x-15$

Answer 1

Factorise the given polynomial $x^2-2x-15$ as shown below:

$x^2-2x-15=0\Rightarrow x^2-5x+3x-15=0\Rightarrow x(x-5)+3(x-5)=0\Rightarrow (x+3)=0,(x-5)=0\Rightarrow x=-3,x=5$

Therefore, the zeroes of the polynomial is $x=-3,5$.

The sum and product of the zeroes is:

Sum$=-3+5=\mathrm{2.........}\left(1\right)$

Product$=(-3)\times \left(5\right)=-\mathrm{15.......}\left(2\right)$

Now, in general, we know that for a equation $a{x}^2+bx+c=0$, if zero are $\alpha$ and $\beta$, plug the values of $a$, $b$ and $c$ we get,

Sum and product of the zeroes is:

Sum$=-\frac{b}{a}=-\frac{-(-2)}{1}=\mathrm{2.......}\left(3\right)$

Product$=\frac{c}{a}=\frac{-15}{1}=-\mathrm{15........}\left(4\right)$

Thus, equation 1 is equal to equation 3 and equation 2 is equal to equation 4.

Hence verified.