Question

How do you solve the equation $2{\left( {x + 2} \right)^2} = 72$?

Answer 1

Hint: We will first divide the given equation by 2. Then we will take the square – root of both the sides. Then, we will subtract 2 from both the sides, so that we have the answer.

Complete step by step solution:
We are given that we are required to solve $2{\left( {x + 2} \right)^2} = 72$.
Dividing both the sides of the above equation by 2, we will then obtain the following equation with us:-
$ \Rightarrow {\left( {x + 2} \right)^2} = 36$
Taking square – root on both the sides of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow x + 2 = \pm 6$
Subtracting 2 from both the sides of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow x + 2 - 2 = \pm 6 - 2$
Simplifying the left hand side of the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow x = \pm 6 - 2$
Simplifying the left hand side of the above mentioned equation further, we will then obtain the following equation with us:-
$ \Rightarrow x = - 8,4$

Note: The students must note that we also have an alternate way to solve the same question but that will be more trickier to do.
Alternate way:
We will first open up the bracket in the left hand side by using the identity given by the following formula:-
$ \Rightarrow {(a + b)^2} = {a^2} + {b^2} + 2ab$
Therefore, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} + 4 + 4x = 36$
Now, taking 26 from addition in the right hand side to subtraction in the left hand side, we will then obtain the following equation:-
$ \Rightarrow {x^2} + 4x - 32 = 0$
Now, we know that if we have the general quadratic equation given by $a{x^2} + bx + c = 0$, then its roots is given by the following equation:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Comparing the above equation, we have: a = 1, b = 4 and c = - 32.
Putting these, we will get:-
$ \Rightarrow x = \dfrac{{ - 4 \pm \sqrt {{4^2} + 4 \times 32} }}{2}$
Simplifying the square – root, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{ - 4 \pm 12}}{2}$
Thus, we have: $x = 4. - 8$.