Solve x=2sqrt{3x-9}

$$x = 2 \sqrt { 3 x - 9 }$$

Solve for x

$$x=6$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(2\sqrt{3x-9}\right)^{2}$$

Expand \left(2\sqrt{3x-9}\right)^{2}.

$$x^{2}=2^{2}\left(\sqrt{3x-9}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}=4\left(\sqrt{3x-9}\right)^{2}$$

Calculate \sqrt{3x-9} to the power of 2 and get 3x-9.

$$x^{2}=4\left(3x-9\right)$$

Use the distributive property to multiply 4 by 3x-9.

$$x^{2}=12x-36$$

Subtract 12x from both sides.

$$x^{2}-12x=-36$$

Add 36 to both sides.

$$x^{2}-12x+36=0$$

To solve the equation, factor x^{2}-12x+36 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-12 ab=36$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -12.

$$a=-6 b=-6$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-6\right)\left(x-6\right)$$

Rewrite as a binomial square.

$$\left(x-6\right)^{2}$$

To find equation solution, solve x-6=0.

$$x=6$$

Substitute 6 for x in the equation x=2\sqrt{3x-9}.

$$6=2\sqrt{3\times 6-9}$$

Simplify. The value x=6 satisfies the equation.

$$6=6$$

Equation x=2\sqrt{3x-9} has a unique solution.

$$x=6$$