Question

How do you solve ${x^2} - x - 12 = 0$ by factoring?

Answer 1

Hint:
Whenever we have given with quadratic equation of the form $a{x^2} + bx + c$ to find the factors, we need to find pair of integers where the product of these integers should be equal to $c$ and the sum of those integers should be $b$. Here by comparing the general quadratic equation with the given question $a = 1$, $b = - 1$ and $c = - 12$. Now finding the pair of integers will lead us to the factors of the given equation.

Complete step by step solution:
Here in this question we have given an equation that is ${x^2} - x - 12 = 0$ which is of the form quadratic equation which represents $a{x^2} + bx + c$ here $a,b,c$ are constants.
Whenever they ask to find the factors, first we need to find a pair of integers where the product of these integers should be equal to $c$ and the sum of those integers should be $b$ in the general quadratic equation.
Now by comparing the given problem statement ${x^2} - x - 12 = 0$ with general form $a{x^2} + bx + c$ we can say that $a = 1$ , $b = - 1$ and $c = - 12$ .
Now we need to find the factors whose product is $ - 12$ and the sum is $ - 1$ .
In this case, we know that $3, - 4$ will give us the required product and the sum that is $3 \times - 4 = - 12 = c$ and $3 - 4 = - 1 = b$ .
Now use these factor in the given equation ${x^2} - x - 12 = 0$ , we get
${x^2} + 3x - 4x - 12 = 0$
We can write the above expression by taking common term as below
$ \Rightarrow x(x + 3) - 4(x + 3) = 0$
$ \Rightarrow (x - 4)(x + 3) = 0$
As the above expression is equated to zero we can write as follows to arrive at the factors of the given equation.
Therefore, $x - 4 = 0$
$ \Rightarrow x = 4$
$x + 3 = 0$
$ \Rightarrow x = - 3$

Therefore the factors of the expression ${x^2} - x - 12 = 0$ is $4, - 3$.

Note:
Whenever solving the quadratic equation you should be careful while finding the pairs of integers which is the major and very important step in the whole procedure, if you fail in this step then you will end up with the wrong answer.