Question

How do you solve ${{\left( 5x-1 \right)}^{2}}=16$ ?

Answer 1

Hint: For answering this question we have been asked to solve the given expression ${{\left( 5x-1 \right)}^{2}}=16$ . For that sake we will apply square root on both sides after that we will have $\Rightarrow 5x-1=\pm 4$ . Now we will further simplify this expression for the solution.

Complete step by step solution:
Now considering the question we have been asked to solve the given expression ${{\left( 5x-1 \right)}^{2}}=16$ .
For that sake we will apply square roots on both sides and simplify the given expression.
After applying square root on both sides of the given expression and simplifying that we will have
$ 5x-1=\pm 4$ .
Now we will further simplify this expression for the solution.
For further simplification we will shift $-1$ from the left hand side of the expression to the right hand side of the expression.
After performing this transformation or shifting on the given expression we will have
$\begin{align}
& \Rightarrow 5x=1\pm 4 \\
& \Rightarrow 5x=5,-3 \\
\end{align}$ .
Now we will shift $5$ from the left hand side of the expression to the right hand side of the expression for further simplifying the expression.
After doing this shifting or transformation on the given expression we will have
$\Rightarrow x=1,\dfrac{-3}{5}$

Therefore we can conclude that the solutions of the given expression ${{\left( 5x-1 \right)}^{2}}=16$ are $1,\dfrac{-3}{5}$.

Note: In the process of answering questions of this type we should be sure with our calculations that we perform during the process. Very few mistakes are possible in this question as it is simple and it can be answered in a short span of time. This question can also be answered by using the formula for finding the roots of the quadratic expression $a{{x}^{2}}+bx+c=0$ which is mathematically given as $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . By using this formula for the expression
 $\begin{align}
  & {{\left( 5x-1 \right)}^{2}}=16\Rightarrow 25{{x}^{2}}-15-10x=0 \\
 & \Rightarrow 5\left( 5{{x}^{2}}-3-2x=0 \right) \\
\end{align}$
We will have
 $\begin{align}
  & \Rightarrow \dfrac{2\pm \sqrt{4-4\left( 5 \right)\left( -3 \right)}}{2\left( 5 \right)}=\dfrac{2\pm \sqrt{4+60}}{10} \\
 & \Rightarrow \dfrac{2\pm \sqrt{64}}{10}=\dfrac{2\pm 8}{10} \\
 & \Rightarrow \dfrac{-3}{5},1 \\
\end{align}$ .
This can be done without using formula also as follows
 $\begin{align}
  & \Rightarrow 5\left( 5{{x}^{2}}-5x+3x-3=0 \right) \\
 & \Rightarrow 5\left( \left( 5x+3 \right)\left( x-1 \right)=0 \right) \\
 & \Rightarrow x=\dfrac{-3}{5},1 \\
\end{align}$ .
We end up with the same answer in all the 3 cases.