Question

How do you solve \[{(x + 2)^2} = 36\]?

Answer 1

Hint:here we need to solve for ‘x’. When we apply the algebraic identity \[{(a + b)^2} = {a^2} + {b^2} + 2ab\] on the left hand side of the equation and simplifying we obtained a polynomial of degree two. That is we have a quadratic equation. Since the degree is two we have two roots or two factors. Here we simply take a square root on the both sides of the equation and on further simplification we obtain a desired result.

Complete step by step solution:
Given
\[{(x + 2)^2} = 36\]
Taking square root on both sides we have,
\[\sqrt {{{(x + 2)}^2}} = \pm \sqrt {36} \]
We know that 36 is a perfect square, then we have
\[\sqrt {{{(x + 2)}^2}} = \pm \sqrt {{6^2}} \]
We know square and square root will cancel out, we have:
\[(x + 2) = \pm 6\]
Thus we have two roots.
\[ \Rightarrow x + 2 = 6\] and \[x + 2 = - 6\]
\[ \Rightarrow x = 6 - 2\] and \[x = - 6 - 2\]
\[ \Rightarrow x = 4\] and \[x = - 8\].
Thus the solutions of \[{(x + 2)^2} = 36\] are \[x = 4\] and \[x = - 8\].

Additional information:
We can solve this using the algebraic identity \[{(a + b)^2} = {a^2} + {b^2} + 2ab\] and then
simplifying we will have a quadratic equation. We solve the quadratic equation using factorization or by using quadratic formula or by completing the square method. It will be a little difficult and will have more steps compared to what we have done above. If we do this we will get the same answer. (That is what we obtained in above.)

Note: We know that roots are also called zeros. We can check whether the obtained result is correct or wrong. All we need to do is substitute the obtained value in the given question.
Now put \[x = 4\]in \[{(x + 2)^2} = 36\],
\[{(4 + 2)^2} = 36\]
\[
{(6)^2} = 36 \\
\Rightarrow 36 = 36 \\
\]
Hence \[x = 4\] is the correct answer.
Now put \[x = - 8\] in \[{(x + 2)^2} = 36\],
\[
{( - 8 + 2)^2} = 36 \\
{( - 6)^2} = 36 \\
\Rightarrow 36 = 36 \\
\]
Hence \[x = - 8\] is the correct answer.