Question

Find the value of x, if:
(i) 4x = (52)² − (48)²
(ii) 14x = (47)² − (33)²
(iii) 5x = (50)² − (40)²

Answer 1

(i) Let us consider the following equation:

$4x={\left(52\right)}^2-{\left(48\right)}^2$
Using the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$, we get:
$$\begin{gathered} 4x={\left(52\right)}^2-{\left(48\right)}^2 \\ 4x=\left(52+48\right)\left(52-48\right) \\ 4x=100\times 4=400 \end{gathered}$$
$\Rightarrow 4x=400$
$\Rightarrow x=100$ (Dividing both sides by 4)

(ii) Let us consider the following equation:

$14x={\left(47\right)}^2-{\left(33\right)}^2$
Using the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$, we get:
$$\begin{gathered} 14x={\left(47\right)}^2-{\left(33\right)}^2 \\ 14x=\left(47+33\right)\left(47-33\right) \\ 14x=80\times 14=1120 \end{gathered}$$
$\Rightarrow 14x=1120$
$\Rightarrow x=80$ (Dividing both sides by 14)

(iii) Let us consider the following equation:

$5x={\left(50\right)}^2-{\left(40\right)}^2$
Using the identity $\left(a+b\right)\left(a-b\right)=a^2-b^2$, we get:
$$\begin{gathered} 5x={\left(50\right)}^2-{\left(40\right)}^2 \\ 5x=\left(50+40\right)\left(50-40\right) \\ 5x=90\times 10=900 \end{gathered}$$
$\Rightarrow 5x=900$
$\Rightarrow x=180$ (Dividing both sides by 5)