Question

How do you solve ${x^2} = 72$?

Answer 1

Hint: To solve the given equation, we first need to determine the degree of the equation. Since the degree of the equation is equal to two, so the given equation is a quadratic equation. Then, we need to write the given quadratic equation into the standard form. Finally, using the quadratic formula and substituting the values of the coefficients into the formula, we will get the required solutions of the given equation.

Formula used:
The formula used to solve this question is given by
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, here $x$ is the solution of a quadratic equation which is written in the form $a{x^2} + bx + c = 0$.

Complete step-by-step solution:
According to the question, we have to solve the equation which is given as
${x^2} = 72$
Since the highest power of the variable $x$ is equal to $2$, so the given equation is a quadratic equation. So for solving it, we first need to write it in the standard form of a quadratic equation. For this, we subtract $72$ from both the sides of the above equation to get
${x^2} - 72 = 0$
Comparing the standard form of the quadratic equation, $a{x^2} + bx + c = 0$, we get $a = 1$, $b = 0$ and $c = - 72$. Now, we know that the solution of a quadratic equation is given by the quadratic formula, which is given by
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Substituting the values of the coefficients, $a = 1$, $b = 0$ and $c = - 72$ in the above formula, we get
$x = \dfrac{{ - 0 \pm \sqrt {{{\left( 0 \right)}^2} - 4 \times 1{\kern 1pt} \times \left( { - 72} \right)} }}{{2 \times 1}}$
$ \Rightarrow x = \pm \dfrac{{2\sqrt {72} }}{2}$
We know that $\sqrt {72} = 6\sqrt 2 $. Putting this above, we get
$ \Rightarrow x = \pm \dfrac{{2\left( {2\sqrt 6 } \right)}}{2}$
$ \Rightarrow x = \pm 2\sqrt 6 $

Hence, the solutions of the given equation are \[x = 2\sqrt 6 \] and \[x = - 2\sqrt 6 \].

Note:
A quadratic equation is an equation that has the highest degree of 2 and has two solutions. There are many types of equations along with quadratic equations such as linear equations, cubic equations, etc. A linear equation is an equation that has the highest degree and has one solution. Also, the cubic equation is an equation that has the highest degree of 3 and has 3 solutions. The number of solutions is equal to the highest degree of the equation.