Solve sqrt{6x+4}+2=x

$$\sqrt { 6 x + 4 } + 2 = x$$

Solve for x

$$x=10$$

Solution Steps

Subtract 2 from both sides of the equation.

$$\sqrt{6x+4}=x-2$$

Square both sides of the equation.

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

Calculate \sqrt{6x+4} to the power of 2 and get 6x+4.

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

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

$$6x+4=x^{2}-4x+4$$

Subtract x^{2} from both sides.

$$6x+4-x^{2}=-4x+4$$

Add 4x to both sides.

$$6x+4-x^{2}+4x=4$$

Combine 6x and 4x to get 10x.

$$10x+4-x^{2}=4$$

Subtract 4 from both sides.

$$10x+4-x^{2}-4=0$$

Subtract 4 from 4 to get 0.

$$10x-x^{2}=0$$

Factor out x.

$$x\left(10-x\right)=0$$

To find equation solutions, solve x=0 and 10-x=0.

$$x=0 x=10$$

Substitute 0 for x in the equation \sqrt{6x+4}+2=x.

$$\sqrt{6\times 0+4}+2=0$$

Simplify. The value x=0 does not satisfy the equation.

$$4=0$$

Substitute 10 for x in the equation \sqrt{6x+4}+2=x.

$$\sqrt{6\times 10+4}+2=10$$

Simplify. The value x=10 satisfies the equation.

$$10=10$$

Equation \sqrt{6x+4}=x-2 has a unique solution.

$$x=10$$