Question

How do you solve by factoring ${x^2} + x - 30 = 0$?

Answer 1

Hint: The problem is based on the factorization method.
Factorization is the process of making factors of the given constant or polynomial having variables and constants.
In this problem we will use the method of splitting the middle term to factorize the given equation.

Complete step-by-step solution:
To factorize the above given quadratic equation we have two methods to solve one is the standard formula method and splitting the middle term, in this problem we will apply splitting the middle term method to factorize the quadratic equation.
In the method of splitting the middle term we have to find two integers, which in addition or subtraction makes 1 and on multiplication it makes 30.
So, let’s check out those two numbers,
10 and 3, 6 and 5 makes 30, out of these pairs only 6 and 5 can give 1 and on multiplication gives 30.
Therefore, we can split the equation as;
$ \Rightarrow {x^2} + x - 30 = 0$
$ \Rightarrow {x^2} + 6x - 5x - 30 = 0$ (We have split the terms)
$ \Rightarrow x(x + 6) - 5(x + 6) = 0$ (We have took out common from the four terms)
$ \Rightarrow \left( {x - 5} \right)\left( {x + 6} \right) = 0$ (We will equate the two terms to 0 in order to find the two factors)
$ \Rightarrow x = 5,x = - 6$

Therefore the values of x are 5 and -6.

Note: There is another method of making the factors of quadratic equation which is the standard one;
$\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ , where a is the coefficient of x2, b is the coefficient of x and c is the constant term. The above formula gives the two factors as the formula contains two signs plus and minus.