Question

How do you factor $4{x^2} - 36?$

Answer 1

Hint: We will first take out 4 common from both the terms in the given binomial. Then we will use the identity given by ${a^2} - {b^2} = (a - b)(a + b)$ to find out the factors.

Complete step by step solution:
We are given that we are required to factor $4{x^2} - 36$.
We see that 36 can be written as a multiple of 4 and 9.
Therefore, we can take out 4 from both the terms and then we can write the given equation as follows:-
$ \Rightarrow 4{x^2} - 36 = 4\left( {{x^2} - 9} \right)$ ……………(1)
Now, we see that we have a formula given by the following expression:-
$ \Rightarrow {a^2} - {b^2} = (a - b)(a + b)$
We will now replace a by x and b by 3 in the above mentioned formula, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} - {3^2} = (x - 3)(x + 3)$
Simplifying the left hand side of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} - 9 = (x - 3)(x + 3)$
Putting this in equation number 1, we will then obtain the following equation with us:-
$ \Rightarrow 4{x^2} - 36 = 4(x - 3)(x + 3)$
Thus, we have the required factors.

Note:-
The students must note that there is an alternate way to do the same:-
Alternate Way 1:
We are given that we are required to factor $4{x^2} - 36$.
We can see that we can write this given equation as follows:-
$ \Rightarrow 4{x^2} - 36 = {\left( {2x} \right)^2} - {6^2}$
Now, we see that we have a formula given by the following expression:-
$ \Rightarrow 4{x^2} - 36 = (4x - 6)(4x + 6)$
Thus, we have obtained the factors.
Alternate Way 2:
We are given that we are required to factor $4{x^2} - 36$.
The general quadratic equation is given by $a{x^2} + bx + c = 0$ whose roots are given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.
Replacing a by 4, b by 0 and c by – 36, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{ - 0 \pm \sqrt {{0^2} - 4 \times 4 \times (36)} }}{{2(4)}}$
Simplifying the above calculations, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{ \pm 4 \times 6}}{8}$
Thus, we have $x = \pm 3$.
Hence, we get the following equation:-
$ \Rightarrow 4{x^2} - 36 = 4(x - 3)(x + 3)$