Question

How do you solve \[{x^2} = 121\]?

Answer 1

Hint: The equation given to us is the quadratic equation. First, we will convert this equation into the standard form. Then, we will calculate the discriminant of the quadratic equation formed using the discriminant formula. We will then substitute the value of discriminant along with the value of coefficients in the quadratic formula to get the final solutions of the given equation.

Formula used:
The formulae used for solving this question are:
\[x = \dfrac{{ - b \pm \sqrt D }}{{2a}}\]
\[D = {b^2} - 4ac\]
Here \[a\], \[b\] and \[c\] are respectively the coefficients of \[{x^2}\], \[x\] and the constant term of the quadratic equation. \[D\] is the discriminant.

Complete step-by-step solution:
The given equation is \[{x^2} = 121\].
Subtracting \[121\] from both the sides, we get
\[{x^2} - 121 = 0\]………………\[\left( 1 \right)\]
Now comparing equation \[\left( 1 \right)\] with the stand quadratic equation \[a{x^2} + bx + c = 0\], we obtain the values of the coefficients as:
\[a = 1,b = 0,c = - 121\]
We know that the discriminant is given by \[D = {b^2} - 4ac\].
Substituting the values of the coefficients in the above formula, we get
\[D = {\left( 0 \right)^2} - 4\left( 1 \right)\left( { - 121} \right)\]
Multiplying the terms, we get
\[ \Rightarrow D = 484\]
Now, we put the values of the coefficients and the discriminant in the formula \[x = \dfrac{{ - b \pm \sqrt D }}{{2a}}\] to get the solutions of the given equation. Therefore, we get
\[x = \dfrac{{ - \left( 0 \right) \pm \sqrt {484} }}{{2\left( 1 \right)}}\]
Simplifying the equation, we get
\[ \Rightarrow x = \pm \dfrac{{22}}{2}\]
Dividing 22 by 2, we get
\[x = \pm 11\]

Hence, the solutions of the given equation are \[x = 11\] and \[x = - 11\].

Note: We can solve the question by using another method.
The given equation is \[{x^2} = 121\].
Subtracting \[121\] from both the sides, we get
\[ \Rightarrow {x^2} - 121 = 0\]
We know that 121 is the square of 11. So we can write above equation as
\[ \Rightarrow {x^2} - {\left( {11} \right)^2} = 0\]
Now using the algebraic identity \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\], we get
\[ \Rightarrow \left( {x + 11} \right)\left( {x - 11} \right) = 0\]
Using the zero product property, we get
\[\begin{array}{l} \Rightarrow \left( {x + 11} \right) = 0\\ \Rightarrow x = - 11\end{array}\]
Or
\[\begin{array}{l} \Rightarrow \left( {x - 11} \right) = 0\\ \Rightarrow x = 11\end{array}\]
Therefore, the solutions of the given equation are \[x = 11\] and \[x = - 11\].