Step 1

Given:

\(\displaystyle{4}\sqrt{{{1}+{3}{x}}}+\sqrt{{{6}{x}+{3}}}=\sqrt{{-{6}{x}-{1}}}\)

Step 2

The objective is to solve the equation

\(\displaystyle{4}\sqrt{{{1}+{3}{x}}}+\sqrt{{{6}{x}+{3}}}=\sqrt{{-{6}{x}-{1}}}\)

On removing the roots

\(\displaystyle{1152}{x}^{{{2}}}+{960}{x}+{192}={3600}{x}^{{{2}}}+{2400}{x}+{400}\)

On solving,

\(\displaystyle{1152}{x}^{{{2}}}+{960}{x}+{192}={3600}{x}^{{{2}}}+{2400}{x}+{400}\)

\(\displaystyle{x}=-{\frac{{{1}}}{{{3}}}},{x}=-{\frac{{{13}}}{{{51}}}}\)

Step 3

On verifying the solution

\(\displaystyle{x}=-{\frac{{{1}}}{{{3}}}}\) it is true

\(\displaystyle{x}=-{\frac{{{13}}}{{{51}}}}\) it is false

Therefore the solution is \(\displaystyle{x}=−{\frac{{{1}}}{{{3}}}}\)

Given:

\(\displaystyle{4}\sqrt{{{1}+{3}{x}}}+\sqrt{{{6}{x}+{3}}}=\sqrt{{-{6}{x}-{1}}}\)

Step 2

The objective is to solve the equation

\(\displaystyle{4}\sqrt{{{1}+{3}{x}}}+\sqrt{{{6}{x}+{3}}}=\sqrt{{-{6}{x}-{1}}}\)

On removing the roots

\(\displaystyle{1152}{x}^{{{2}}}+{960}{x}+{192}={3600}{x}^{{{2}}}+{2400}{x}+{400}\)

On solving,

\(\displaystyle{1152}{x}^{{{2}}}+{960}{x}+{192}={3600}{x}^{{{2}}}+{2400}{x}+{400}\)

\(\displaystyle{x}=-{\frac{{{1}}}{{{3}}}},{x}=-{\frac{{{13}}}{{{51}}}}\)

Step 3

On verifying the solution

\(\displaystyle{x}=-{\frac{{{1}}}{{{3}}}}\) it is true

\(\displaystyle{x}=-{\frac{{{13}}}{{{51}}}}\) it is false

Therefore the solution is \(\displaystyle{x}=−{\frac{{{1}}}{{{3}}}}\)