Question

How do you factorize ${x^2} + 3x - 18 = 0$ ?

Answer 1

Hint: The above problem is based on the factorization of quadratic equations.
Quadratic equation is the one which has the degree of a variable as 2.
Factors are the multiples or the prime numbers which on being multiplied gives the same number which we have factorized.
Using the method of factorization (splitting the middle term) we will solve the problem.

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.
$ \Rightarrow {x^2} + 3x - 18 = 0$
By splitting the middle term we mean we require two numbers which add up or subtracts to make sum as 3 and on multiplication it gives 18.
The Numbers which have multiplication products as 18 are 3 and 6, 2 and 9, but by adding or subtracting 3 and 6 we can make 3.
$ \Rightarrow {x^2} + 6x - 3x - 18 = 0$ (We will take out common from the four terms)
$ \Rightarrow x(x + 6) - 3(x + 6) = 0$ (While taking out common form the terms we must keep the signs of minus in mind)
$ \Rightarrow \left( {x - 3} \right)\left( {x + 6} \right) = 0$ (On keeping the two brackets equal to 0 we will get two different values of x)
$ \Rightarrow x = 3,x = - 6$

Therefore the values of x are 3 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 $x^2$, 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.