Question

Factor the quadratic expression1.x^2 + 5x - 142.x^2 + 7x + 6

Answer 1

A quadratic equation can be rewritten as

$ax^2+bx+c=a(x-x_1)(x-x_2)$

where x1 and x2 represents the roots of the quadratic equation.

To find those roots we can use the quadratic equation. Given a quadratic equation with the following form

$ax^2+bx+c=0$

its roots are given by

$x_{\pm}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

In our first quadratic equation, we have

$x^2+5x-14$

therefore, its roots are

$\begin{gathered} x_{\pm}=\frac{-(5)\pm\sqrt{(5)^2-4(1)(-14)}}{2(1)} \\ =\frac{-5\pm\sqrt{25+56}}{2} \\ =\frac{-5\pm9}{2} \\ \implies\begin{cases}x_-={-7} \\ x_+={2}\end{cases} \end{gathered}$

Then, this quadratic expression can be factorized as

$x^2+5x-14=(x+7)(x-2)$

Using the same process for the other expression, we have

$x^2+7x+6=(x+1)(x+6)$