Solve 13-sqrt{13+4x}=2sqrt{x}

$$13 - \sqrt { 13 + 4 x } = 2 \sqrt { x }$$

Solve for x

$$x=9$$

Solution Steps

Square both sides of the equation.

$$\left(13-\sqrt{13+4x}\right)^{2}=\left(2\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13-\sqrt{13+4x}\right)^{2}.

$$169-26\sqrt{13+4x}+\left(\sqrt{13+4x}\right)^{2}=\left(2\sqrt{x}\right)^{2}$$

Calculate \sqrt{13+4x} to the power of 2 and get 13+4x.

$$169-26\sqrt{13+4x}+13+4x=\left(2\sqrt{x}\right)^{2}$$

Add 169 and 13 to get 182.

$$182-26\sqrt{13+4x}+4x=\left(2\sqrt{x}\right)^{2}$$

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

$$182-26\sqrt{13+4x}+4x=2^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$182-26\sqrt{13+4x}+4x=4\left(\sqrt{x}\right)^{2}$$

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

$$182-26\sqrt{13+4x}+4x=4x$$

Subtract 4x from both sides.

$$182-26\sqrt{13+4x}+4x-4x=0$$

Combine 4x and -4x to get 0.

$$182-26\sqrt{13+4x}=0$$

Subtract 182 from both sides. Anything subtracted from zero gives its negation.

$$-26\sqrt{13+4x}=-182$$

Divide both sides by -26.

$$\sqrt{13+4x}=\frac{-182}{-26}$$

Divide -182 by -26 to get 7.

$$\sqrt{13+4x}=7$$

Square both sides of the equation.

$$4x+13=49$$

Subtract 13 from both sides of the equation.

$$4x+13-13=49-13$$

Subtracting 13 from itself leaves 0.

$$4x=49-13$$

Subtract 13 from 49.

$$4x=36$$

Divide both sides by 4.

$$\frac{4x}{4}=\frac{36}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x=\frac{36}{4}$$

Divide 36 by 4.

$$x=9$$

Substitute 9 for x in the equation 13-\sqrt{13+4x}=2\sqrt{x}.

$$13-\sqrt{13+4\times 9}=2\sqrt{9}$$

Simplify. The value x=9 satisfies the equation.

$$6=6$$

Equation -\sqrt{4x+13}+13=2\sqrt{x} has a unique solution.

$$x=9$$