Question

How do you solve the equation ${x^2} - 324 = 0$?

Answer 1

Hint: According to the question given in the question we have to determine the solution of the quadratic expression which is ${x^2} - 324 = 0$. So, to solve the given quadratic expression which is ${x^2} - 324 = 0$ first of all we have to make the constant term in the form of square and the constant term in the given quadratic expression is 324.
Now, we have to solve the expression as obtained after making in the form of square with the help of the formula which is as mentioned below:

Formula used: $ \Rightarrow ({a^2} - {b^2}) = (a + b)(a - b)...............(A)$
Hence, with the help of the formula (A) above, we can determine the solution of the expression.
Now, we will obtain two roots of the expression and now we have to compare both of the obtained roots with 0 to obtain the both of the values of x.

Complete step-by-step solution:
Step1: First of all we have to make the constant term in the form of square and the constant term in the given quadratic expression is 324 as mentioned in the solution hint. Hence,
$
   \Rightarrow {x^2} - (18 \times 18) = 0 \\
   \Rightarrow {x^2} - {18^2} = 0..............(1)
 $
Step 2: Now, we have to solve the expression (1) as obtained in the solution step 1 after making it in the form of a square with the help of the formula which is as mentioned in the solution hint. Hence,
$ \Rightarrow (x + 18)(x - 18) = 0............(2)$
Step 3: Now, we will obtain two roots of the expression and now we have to compare both of the obtained roots with 0 to obtain the both of the values of x which is as mentioned in the solution hint. Hence,
$
   \Rightarrow (x + 18) = 0 \\
   \Rightarrow x = - 18
 $
And,
$
   \Rightarrow (x - 18) = 0 \\
   \Rightarrow x = 18
 $

Hence, with the help of the formula (A) we have determined the solution of the given expression ${x^2} - 324 = 0$ is $x = \pm 18$.

Note: It is necessary that we have to make both of the given terms of the expression in the form of square so that we can easily obtain the given expression in the form of ${a^2} - {b^2}$.
On solving the quadratic expression only two roots or zeros can be formed and these both of the obtained roots or zeroes will satisfy the expression as given in the question.