Question

Find the zeros of the quadratic polynomial $x^2+5x+6$ and verify the relationship between the zeros and the coefficients.

Answer 1

Given $x^2+5x+6=0$

Split the middle term $5x$ as sum of two terms $2x$ and $3x$ such that its product $2x\times 3x=6x^2$ is equal to product of extreme terms($x^2\times 6$) .
$\Rightarrow x^2+2x+3x+6=0$

$\Rightarrow x(x+2)+3(x+3)=0$

$\Rightarrow (x+2)(x+3)=0$

$\Rightarrow x+2=0$ and $x+3=0$

$\therefore x=-2,-3$

Let the roots be $\alpha$ and $\beta$, So $\alpha =-2,\beta =-3$.

Since, we have $\alpha +\beta =-\frac{b}{a}$

$\Rightarrow -2-3=-\frac{5}{1}$

$\therefore -5=-5$

And also we have $\alpha \beta =\frac{c}{a}$

$\Rightarrow (-2)(-3)=\frac{6}{1}$

$\therefore 6=6$