Question

Find the value of $x$, if $4x=\left(52\right)^2-\left(48\right)^2$.

Answer 1

Hint: We are given an equation and we have to find the unknown here. We see that the numbers inside the brackets are quite large, so we won’t open the squares and then do the subtraction. We instead will use an algebraic identity which will make our calculations easier. To figure out what identity to use, observe carefully the right side of the equation which involves squaring and subtraction, so we will use the identity $a^2-b^2=\left(a+b\right)\left(a-b\right)$.

Complete step by step answer:
We have $4x=\left(52\right)^2-\left(48\right)^2$. Look at the right side of this equation. We know the following algebraic identity:
For any two real numbers $a$ and $b$, the following holds true in any case:
$a^2-b^2=\left(a+b\right)\left(a-b\right)$
We assign the numbers $a$ and $b$ as follows:
$a=52$ and $b=48$
Then:
$52^2-48^2=\left(52+48\right)\left(52-48\right)$
So, we get:
$52^2-48^2=100\times 4=400$
Now, this result obtained is equal to $4x$ which is the left hand side of the equation given in the question, i.e.
$4x=400$
Divide both sides of the equation by 4 we get:
$x=100$
So, we have found the value of $x$ to be 100.

Note: See that we have reduced the calculations by using a trick identity. Do not open the squares because that would lead to a very large calculation. Also, there is a chance that you might end up making huge calculation mistakes which might lead to an incorrect answer. In such questions, always try to use some algebraic identity wherever possible because that reduces the calculation labor and also gives the most accurate result.