Question

How do you solve ${x^2} - 121 = 0$?

Answer 1

In this question, we need to find the solution for the given equation. Note that the given expression is a quadratic equation. We solve for the unknown variable x. Firstly we will move all the constant terms in the left hand side to the right hand side by adding 121 to both sides of the equation. Then we take square roots on both sides. After that we simplify the expression to get the value for the variable x. Here we get two solutions for the given equation.

Complete step by step answer:
Given the equation ${x^2} - 121 = 0$ …… (1)
We are asked to solve the above equation given in the equation (1). i.e. we need to find the value of the unknown variable x.
Note that the above equation is a quadratic equation of degree two.
We will simplify the given equation and solve for x.
Firstly, let us move all the terms not containing the x to the right hand side.
We do this, by adding 121 to each of the sides of the equation (1), we get,
$\Rightarrow {x^2} - 121 + 121 = 0 + 121$
Combining the like terms and simplifying we get,
$\Rightarrow {x^2} + 0 = 121$
$\Rightarrow {x^2} = 121$
Now taking square root on both sides we get,
$\Rightarrow \sqrt {{x^2}} = \sqrt {121}$
We know the value of $\sqrt {121} = 11$.
Hence we get,
$\Rightarrow \sqrt {{x^2}} = 11$
$\Rightarrow x = \pm 11$

Thus, the solution for the given equation ${x^2} - 121 = 0$ is given by $x = \pm 11$.

Note: Alternative method :
Consider the given equation ${x^2} - 121 = 0$
We know that ${11^2} = 121$
Note that the above equation can also be written as,
$\Rightarrow {x^2} - {11^2} = 0$
This is of the form ${a^2} - {b^2}$.
We have the formula, ${a^2} - {b^2} = (a - b)(a + b)$
Here $a = x$ and $b = 11$.
Hence we get,
$\Rightarrow (x - 11)(x + 11) = 0$
$\Rightarrow x - 11 = 0$ and $x + 11 = 0$
$\Rightarrow x = 11$ and $x = - 11$.
Hence we get, $x = \pm 11$.
Thus, the solution for the given equation ${x^2} - 121 = 0$ is given by $x = \pm 11$.