Solve sqrt{12x+1}=1+sqrt{10x}

$$\sqrt { 12 x + 1 } = 1 + \sqrt { 10 x }$$

Solve for x

$$x=10$$

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{12x+1}\right)^{2}=\left(1+\sqrt{10x}\right)^{2}$$

Calculate \sqrt{12x+1} to the power of 2 and get 12x+1.

$$12x+1=\left(1+\sqrt{10x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{10x}\right)^{2}.

$$12x+1=1+2\sqrt{10x}+\left(\sqrt{10x}\right)^{2}$$

Calculate \sqrt{10x} to the power of 2 and get 10x.

$$12x+1=1+2\sqrt{10x}+10x$$

Subtract 1+10x from both sides of the equation.

$$12x+1-\left(1+10x\right)=2\sqrt{10x}$$

To find the opposite of 1+10x, find the opposite of each term.

$$12x+1-1-10x=2\sqrt{10x}$$

Subtract 1 from 1 to get 0.

$$12x-10x=2\sqrt{10x}$$

Combine 12x and -10x to get 2x.

$$2x=2\sqrt{10x}$$

Cancel out 2 on both sides.

$$x=\sqrt{10x}$$

Square both sides of the equation.

$$x^{2}=\left(\sqrt{10x}\right)^{2}$$

Calculate \sqrt{10x} to the power of 2 and get 10x.

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

Subtract 10x from both sides.

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

Factor out x.

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

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

$$x=0 x=10$$

Substitute 0 for x in the equation \sqrt{12x+1}=1+\sqrt{10x}.

$$\sqrt{12\times 0+1}=1+\sqrt{10\times 0}$$

Simplify. The value x=0 satisfies the equation.

$$1=1$$

Substitute 10 for x in the equation \sqrt{12x+1}=1+\sqrt{10x}.

$$\sqrt{12\times 10+1}=1+\sqrt{10\times 10}$$

Simplify. The value x=10 satisfies the equation.

$$11=11$$

List all solutions of \sqrt{12x+1}=\sqrt{10x}+1.

$$x=0 x=10$$