Question

How do you solve ${{\left( 5x-1 \right)}^{2}}=\dfrac{4}{25}$ ?

Answer 1

In order to find the solution to this problem, we will solve according to ${{\left( g\left( x \right) \right)}^{2}}=f\left( a \right)$ format, so we will use formula: $g\left( x \right)=\sqrt{f\left( a \right)},-\sqrt{f\left( a \right)}$ , simplify it and find the value of $x$ accordingly.

Complete step-by-step solution:
We have our equation:
${{\left( 5x-1 \right)}^{2}}=\dfrac{4}{25}$
From the above problem as we can see that it is in the form of ${{\left( g\left( x \right) \right)}^{2}}=f\left( a \right)$ form.
So we will use formula,
$g\left( x \right)=\sqrt{f\left( a \right)},-\sqrt{f\left( a \right)}$
Therefore, evaluate the above formula in our equation, we get:
$5x-1=\sqrt{\dfrac{4}{25}}$
Here we have used the first part of the formula, that is the positive root of the element.
Now, add 1 to both sides, we get:
$5x-1+1=\sqrt{\dfrac{4}{25}}+1$
First let us simplify left hand side:
$\Rightarrow 5x-1+1$
Now on simplifying and by using similar element property, $-1+1=0$
Therefore, we get:
$\Rightarrow 5x$
Now let us simplify right hand side:
$\Rightarrow \sqrt{\dfrac{4}{25}}+1$
On taking out square roots, we get:
$\Rightarrow \dfrac{2}{5}+1$
Now we will convert the element into fractions.
Therefore, we get:
$\Rightarrow \dfrac{2}{5}+\dfrac{5}{5}$
On adding, we get:
$\Rightarrow \dfrac{7}{5}$
Now we will write both right hand side and left hand side:
$5x=\dfrac{7}{5}$
On simplifying:
$x=\dfrac{7}{5\times 5}$
Therefore, we get:
$x=\dfrac{7}{25}$

Now, proceeding to our second part of formula, we get:
$5x-1=-\sqrt{\dfrac{4}{25}}$
Now, add 1 to both sides, we get:
$5x-1+1=-\sqrt{\dfrac{4}{25}}+1$
First let us simplify left hand side:
$\Rightarrow 5x-1+1$
Now on simplifying and by using similar element property, $-1+1=0$
Therefore, we get:
$\Rightarrow 5x$
Now let us simplify right hand side:
$\Rightarrow -\sqrt{\dfrac{4}{25}}+1$
On taking out square roots, we get:
$\Rightarrow -\dfrac{2}{5}+1$
Now we will convert the element into fractions.
Therefore, we get:
$\Rightarrow -\dfrac{2}{5}+\dfrac{5}{5}$
On simplifying, we get:
$\Rightarrow \dfrac{3}{5}$
Now we will write both right hand side and left hand side:
$5x=\dfrac{3}{5}$
On simplifying:
$x=\dfrac{3}{5\times 5}$
Therefore, we get:
$x=\dfrac{3}{25}$
Finally as we can see that we have the value of $x$, that is:
$x=\dfrac{7}{25}$ and $x=\dfrac{3}{25}$.

Note: To find whether the value of $x$ is correct, we can substitute it in the given equation and equate it.
${{\left( 5x-1 \right)}^{2}}=\dfrac{4}{25}$
On substituting $x=\dfrac{7}{25}$ in the left-hand side we get:
$\Rightarrow {{\left( 5\times \dfrac{7}{25}-1 \right)}^{2}}$
Now by applying $BODMAS$ rule, we get:
$\Rightarrow {{\left( \dfrac{7}{5}-1 \right)}^{2}}$
On simplifying and squaring, we get:
$\Rightarrow {{\left( \dfrac{7}{5}-\dfrac{5}{5} \right)}^{2}}$
$\Rightarrow {{\left( \dfrac{7-5}{5} \right)}^{2}}$
$\Rightarrow {{\left( \dfrac{2}{5} \right)}^{2}}$
$\Rightarrow \dfrac{4}{25}$
${{\left( 5x-1 \right)}^{2}}=\dfrac{4}{25}$
Since the left-hand side equals to the right-hand side, we can conclude that the answer is correct.