Question

How do you solve $3{{x}^{2}}=108$?

Answer 1

Hint: We will see that the terms on both sides of the equation are divisible by three. We will simplify the equation by dividing both sides by three. Then we will rearrange the equation to make it look like ${{x}^{2}}-{{a}^{2}}=0$. After that we can use the identity ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$ to factorize the obtained equation and obtain the possible values of the variable.

Complete answer:
The given equation is $3{{x}^{2}}=108$. We can see that the left hand side of the equation has 3 as a factor. On the right hand side, we have the number 108. Using the divisibility test for 3, we can see that the number 108 is divisible by 3. Dividing 108 by 3, we get $108\div 3=36$. Let us divide the given equation by the number 3. So, we have the following,
\[\begin{align}
  & \dfrac{3{{x}^{2}}}{3}=\dfrac{108}{3} \\
 & \therefore {{x}^{2}}=36 \\
\end{align}\]
Now, we will shift the constant term to the left hand side in the following manner,
${{x}^{2}}-36=0$
We have an identity that factorizes equations of this type. The identity is given as ${{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)$. Using this identity, we will factorize the above equation. So, we get
$\left( x-6 \right)\left( x+6 \right)=0$
Therefore, we have either $x-6=0$ or $x+6=0$. Hence, we obtain the value of the variable as $x=6$ or $x=-6$.

Note: The divisibility test for three is to check the sum of the digits of the number to be divided. If the sum of the digits is a multiple of 3, then the number is divisible by three. There are divisibility tests for other numbers as well. These tests are useful in doing calculations. Factorization of a quadratic equation is one method of solving the equation. We have other methods like the quadratic formula or completing square method as well.