Solve sqrt{6x+1}=2x+1

$$\sqrt { 6 x + 1 } = 2 x + 1$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{6x+1}\right)^{2}=\left(2x+1\right)^{2}$$

Calculate \sqrt{6x+1} to the power of 2 and get 6x+1.

$$6x+1=\left(2x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+1\right)^{2}.

$$6x+1=4x^{2}+4x+1$$

Subtract 4x^{2} from both sides.

$$6x+1-4x^{2}=4x+1$$

Subtract 4x from both sides.

$$6x+1-4x^{2}-4x=1$$

Combine 6x and -4x to get 2x.

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

Subtract 1 from both sides.

$$2x+1-4x^{2}-1=0$$

Subtract 1 from 1 to get 0.

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

Factor out x.

$$x\left(2-4x\right)=0$$

To find equation solutions, solve x=0 and 2-4x=0.

$$x=0 x=\frac{1}{2}$$

Substitute 0 for x in the equation \sqrt{6x+1}=2x+1.

$$\sqrt{6\times 0+1}=2\times 0+1$$

Simplify. The value x=0 satisfies the equation.

$$1=1$$

Substitute \frac{1}{2} for x in the equation \sqrt{6x+1}=2x+1.

$$\sqrt{6\times \frac{1}{2}+1}=2\times \frac{1}{2}+1$$

Simplify. The value x=\frac{1}{2} satisfies the equation.

$$2=2$$

List all solutions of \sqrt{6x+1}=2x+1.

$$x=0 x=\frac{1}{2}$$