Question

Find the zero of the quadratic polynomial $x^2+7x+10$, and verify the relationship between the zeros and coefficients.

Answer 1

Let $p\left(x\right)=x^2+7x+10$ comparing with $p\left(x\right)=a{x}^2+bx+c$, we get

$a=1,b=7,c=10$

$p\left(x\right)=x^2+7x+10$

$=x^2+5x+2x+10$

$=x(x+5)+2(x+5)$

$=(x+2)(x+5)$

Let $p\left(x\right)=0$
$\Rightarrow (x+2)(x+5)=0$

$x=-2,5$

Let $\alpha =-2,\beta =-5$
Since, $\frac{-b}{a}=\alpha +\beta \Rightarrow \frac{-\left(7\right)}{1}=-5-2=-7$
$\frac{c}{a}=\alpha \cdot \beta \Rightarrow \frac{10}{1}=(-2)\times (-5)=10$