# Calculate: sqrt(3x)-sqrt(x+4)=2

## Expression: $\sqrt{ 3x }-\sqrt{ x+4 }=2$

Square both sides of the equation

$3x-2\sqrt{ 3x \times \left( x+4 \right) }+x+4=4$

Cancel equal terms on both sides of the equation

$3x-2\sqrt{ 3x \times \left( x+4 \right) }+x=0$

Distribute $3x$ through the parentheses

$3x-2\sqrt{ 3{x}^{2}+12x }+x=0$

Collect like terms

$4x-2\sqrt{ 3{x}^{2}+12x }=0$

Move the variable to the right-hand side and change its sign

$-2\sqrt{ 3{x}^{2}+12x }=-4x$

Divide both sides of the equation by $-2$

$\sqrt{ 3{x}^{2}+12x }=2x$

Square both sides of the equation

$3{x}^{2}+12x=4{x}^{2}$

Move the variable to the left-hand side and change its sign

$3{x}^{2}+12x-4{x}^{2}=0$

Collect like terms

$-{x}^{2}+12x=0$

Factor out $-x$ from the expression

$-x \times \left( x-12 \right)=0$

Change the signs on both sides of the equation

$x \times \left( x-12 \right)=0$

When the product of factors equals $0$, at least one factor is $0$

$\begin{array} { l }x=0,\\x-12=0\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=0,\\x=12\end{array}$

Check if the given value is the solution of the equation

$\begin{array} { l }\sqrt{ 3 \times 0 }-\sqrt{ 0+4 }=2,\\x=12\end{array}$

Check if the given value is the solution of the equation

$\begin{array} { l }\sqrt{ 3 \times 0 }-\sqrt{ 0+4 }=2,\\\sqrt{ 3 \times 12 }-\sqrt{ 12+4 }=2\end{array}$

Simplify the expression

$\begin{array} { l }-2=2,\\\sqrt{ 3 \times 12 }-\sqrt{ 12+4 }=2\end{array}$

Simplify the expression

$\begin{array} { l }-2=2,\\2=2\end{array}$

The equality is false, therefore $x=0$ is not a solution of the equation

$\begin{array} { l }x≠0,\\2=2\end{array}$

The equality is true, therefore $x=12$ is a solution of the equation

$\begin{array} { l }x≠0,\\x=12\end{array}$

The equation has one solution

$x=12$

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