Solve the equation. $\newline$ $\sqrt{5 x+1}-1=\sqrt{3 x}$

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Q. Solve the equation.

\newline

\sqrt{5 x+1}-1=\sqrt{3 x}

Isolate square root: Isolate one of the square roots. $\sqrt{5x+1} = \sqrt{3x} + 1$
Square both sides: Square both sides to eliminate the square root on the left side. $\newline$ (\sqrt{$5$x+$1$})^\(2 = (\sqrt{ $3$ x} + $1$ )^ $2$
Apply squaring operation: Apply the squaring operation to both sides. $\newline$ $5x + 1 = (\sqrt{3x})^2 + 2\cdot\sqrt{3x}\cdot1 + 1^2$
Simplify right side: Simplify the right side of the equation. $5x + 1 = 3x + 2\sqrt{3x} + 1$
Subtract to isolate: Subtract $3x$ and $1$ from both sides to isolate the square root term. $\newline$ $5x + 1 - 3x - 1 = 2\sqrt{3x}$
Combine like terms: Combine like terms. $2x = 2\sqrt{3x}$
Divide to isolate: Divide both sides by $2$ to isolate the square root. $x = \sqrt{3x}$
Square both sides again: Square both sides again to eliminate the square root. $x^2 = (\sqrt{3x})^2$
Apply squaring operation: Apply the squaring operation to both sides. $\newline$ $x^2 = 3x$
Set equation to zero: Subtract $3x$ from both sides to set the equation to zero. $\newline$ $x^2 - 3x = 0$
Factor out $x$ : Factor out $x$ from the left side of the equation. $x(x - 3) = 0$
Apply zero product property: Apply the zero product property. $x = 0$ or $x - 3 = 0$
Solve for x: Solve for x. $\newline$ $x = 0$ or $x = 3$
Check solutions: Check the solutions in the original equation to ensure they do not produce a negative number under the square root or an incorrect equality. $\newline$ For $x = 0$ : $\newline$ $\sqrt{5(0)+1}-1 = \sqrt{3(0)}$ $\newline$ $\sqrt{1}-1 = \sqrt{0}$ $\newline$ $1 - 1 = 0$ $\newline$ $0 = 0$ (Valid solution) $\newline$ For $x = 3$ : $\newline$ $\sqrt{5(3)+1}-1 = \sqrt{3(3)}$ $\newline$ $\sqrt{16}-1 = \sqrt{9}$ $\newline$ $4 - 1 = 3$ $\newline$ $3 = 3$ (Valid solution) $\newline$ Both solutions are valid.