Answer

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**Hint:**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 $x=\dfrac{3}{25}$.

$x=\dfrac{7}{25}$ 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.

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