Question

How do you factor ${x^2} + 8x - 19 = 0$?

Answer 1

Hint: Given a quadratic equation. We have to factor the equation given in the form of $a{x^2} + bx + c = 0$. First, we will check whether it is possible to split the middle term of the equation whose sum is equal to $b$ and product is equal to $a \times c$. If possible, then factorize the equation. If it is not possible to factorize the equation, then apply the quadratic formula to find the factors of the equation.

Formula used:
The quadratic formula for the quadratic equation $a{x^2} + bx + c = 0$ is given by:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Complete step-by-step answer:
We are given the equation ${x^2} + 8x - 19 = 0$. Here the value of $a = 1$, $b = 8$ and $c = - 19$. First, we will try to split the middle term whose sum is $8$ and product is $ - 19$.
Find the factors of the term $19$.
$ \Rightarrow 1 \times 19$
Using $1$ and $19$, the sum $8$ is not possible.
Thus, factorization of the equation by splitting the middle term is not possible.
Now, we will apply the quadratic formula by substituting $a = 1$, $b = 8$ and $c = - 19$.
$ \Rightarrow x = \dfrac{{ - 8 \pm \sqrt {{8^2} - 4\left( 1 \right)\left( { - 19} \right)} }}{{2 \times 1}}$
On simplifying the expression, we get:
$ \Rightarrow x = \dfrac{{ - 8 \pm \sqrt {64 + 76} }}{2}$
$ \Rightarrow x = \dfrac{{ - 8 \pm \sqrt {140} }}{2}$
Now, find the factors of the radicand $140$.
$ \Rightarrow x = \dfrac{{ - 8 \pm \sqrt {4 \times 35} }}{2}$
Take out the perfect square from the radicand.
$ \Rightarrow x = \dfrac{{ - 8 \pm 2\sqrt {35} }}{2}$
Now, divide the expression by $2$.
$ \Rightarrow x = - 4 \pm \sqrt {35} $

Thus, the factors of the polynomial are $x = - 4 + \sqrt {35} $ and $x = - 4 - \sqrt {35} $

Note:
In such types of questions students mainly don't get an approach on how to solve it. In such types of questions students mainly forget to apply the middle term splitting to check whether the factorization is possible or not. Students may get confused on which method is used to find the factors of the equation if middle term splitting is not possible.