Solve sqrt{x}+sqrt{x+6}=sqrt{6+x}

$$\sqrt { x } + \sqrt { x + 6 } = \sqrt { 6 + x }$$

Solve for x

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}+\sqrt{x+6}\right)^{2}=\left(\sqrt{6+x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x}+\sqrt{x+6}\right)^{2}.

$$\left(\sqrt{x}\right)^{2}+2\sqrt{x}\sqrt{x+6}+\left(\sqrt{x+6}\right)^{2}=\left(\sqrt{6+x}\right)^{2}$$

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

$$x+2\sqrt{x}\sqrt{x+6}+\left(\sqrt{x+6}\right)^{2}=\left(\sqrt{6+x}\right)^{2}$$

Calculate \sqrt{x+6} to the power of 2 and get x+6.

$$x+2\sqrt{x}\sqrt{x+6}+x+6=\left(\sqrt{6+x}\right)^{2}$$

Combine x and x to get 2x.

$$2x+2\sqrt{x}\sqrt{x+6}+6=\left(\sqrt{6+x}\right)^{2}$$

Calculate \sqrt{6+x} to the power of 2 and get 6+x.

$$2x+2\sqrt{x}\sqrt{x+6}+6=6+x$$

Subtract 2x+6 from both sides of the equation.

$$2\sqrt{x}\sqrt{x+6}=6+x-\left(2x+6\right)$$

To find the opposite of 2x+6, find the opposite of each term.

$$2\sqrt{x}\sqrt{x+6}=6+x-2x-6$$

Combine x and -2x to get -x.

$$2\sqrt{x}\sqrt{x+6}=6-x-6$$

Subtract 6 from 6 to get 0.

$$2\sqrt{x}\sqrt{x+6}=-x$$

Square both sides of the equation.

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

Expand \left(2\sqrt{x}\sqrt{x+6}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x}\right)^{2}\left(\sqrt{x+6}\right)^{2}=\left(-x\right)^{2}$$

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

$$4x\left(\sqrt{x+6}\right)^{2}=\left(-x\right)^{2}$$

Calculate \sqrt{x+6} to the power of 2 and get x+6.

$$4x\left(x+6\right)=\left(-x\right)^{2}$$

Use the distributive property to multiply 4x by x+6.

$$4x^{2}+24x=\left(-x\right)^{2}$$

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

$$4x^{2}+24x=\left(-1\right)^{2}x^{2}$$

Calculate -1 to the power of 2 and get 1.

$$4x^{2}+24x=1x^{2}$$

Reorder the terms.

$$4x^{2}+24x=x^{2}$$

Subtract x^{2} from both sides.

$$4x^{2}+24x-x^{2}=0$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+24x=0$$

Factor out x.

$$x\left(3x+24\right)=0$$

To find equation solutions, solve x=0 and 3x+24=0.

$$x=0 x=-8$$

Substitute -8 for x in the equation \sqrt{x}+\sqrt{x+6}=\sqrt{6+x}. The expression \sqrt{-8} is undefined because the radicand cannot be negative.

$$\sqrt{-8}+\sqrt{-8+6}=\sqrt{6-8}$$

Substitute 0 for x in the equation \sqrt{x}+\sqrt{x+6}=\sqrt{6+x}.

$$\sqrt{0}+\sqrt{0+6}=\sqrt{6+0}$$

Simplify. The value x=0 satisfies the equation.

$$6^{\frac{1}{2}}=6^{\frac{1}{2}}$$

Equation \sqrt{x+6}+\sqrt{x}=\sqrt{x+6} has a unique solution.

$$x=0$$