Solve sqrt{10x+1}-sqrt{2x}=sqrt{3x+1}

$$\sqrt { 10 x + 1 } - \sqrt { 2 x } = \sqrt { 3 x + 1 }$$

Solve for x

$$x=8$$

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{10x+1}-\sqrt{2x}\right)^{2}=\left(\sqrt{3x+1}\right)^{2}$$

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

$$\left(\sqrt{10x+1}\right)^{2}-2\sqrt{10x+1}\sqrt{2x}+\left(\sqrt{2x}\right)^{2}=\left(\sqrt{3x+1}\right)^{2}$$

Calculate \sqrt{10x+1} to the power of 2 and get 10x+1.

$$10x+1-2\sqrt{10x+1}\sqrt{2x}+\left(\sqrt{2x}\right)^{2}=\left(\sqrt{3x+1}\right)^{2}$$

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

$$10x+1-2\sqrt{10x+1}\sqrt{2x}+2x=\left(\sqrt{3x+1}\right)^{2}$$

Combine 10x and 2x to get 12x.

$$12x+1-2\sqrt{10x+1}\sqrt{2x}=\left(\sqrt{3x+1}\right)^{2}$$

Calculate \sqrt{3x+1} to the power of 2 and get 3x+1.

$$12x+1-2\sqrt{10x+1}\sqrt{2x}=3x+1$$

Subtract 12x+1 from both sides of the equation.

$$-2\sqrt{10x+1}\sqrt{2x}=3x+1-\left(12x+1\right)$$

To find the opposite of 12x+1, find the opposite of each term.

$$-2\sqrt{10x+1}\sqrt{2x}=3x+1-12x-1$$

Combine 3x and -12x to get -9x.

$$-2\sqrt{10x+1}\sqrt{2x}=-9x+1-1$$

Subtract 1 from 1 to get 0.

$$-2\sqrt{10x+1}\sqrt{2x}=-9x$$

Square both sides of the equation.

$$\left(-2\sqrt{10x+1}\sqrt{2x}\right)^{2}=\left(-9x\right)^{2}$$

Expand \left(-2\sqrt{10x+1}\sqrt{2x}\right)^{2}.

$$\left(-2\right)^{2}\left(\sqrt{10x+1}\right)^{2}\left(\sqrt{2x}\right)^{2}=\left(-9x\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4\left(\sqrt{10x+1}\right)^{2}\left(\sqrt{2x}\right)^{2}=\left(-9x\right)^{2}$$

Calculate \sqrt{10x+1} to the power of 2 and get 10x+1.

$$4\left(10x+1\right)\left(\sqrt{2x}\right)^{2}=\left(-9x\right)^{2}$$

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

$$4\left(10x+1\right)\times 2x=\left(-9x\right)^{2}$$

Multiply 4 and 2 to get 8.

$$8\left(10x+1\right)x=\left(-9x\right)^{2}$$

Use the distributive property to multiply 8 by 10x+1.

$$\left(80x+8\right)x=\left(-9x\right)^{2}$$

Use the distributive property to multiply 80x+8 by x.

$$80x^{2}+8x=\left(-9x\right)^{2}$$

Expand \left(-9x\right)^{2}.

$$80x^{2}+8x=\left(-9\right)^{2}x^{2}$$

Calculate -9 to the power of 2 and get 81.

$$80x^{2}+8x=81x^{2}$$

Subtract 81x^{2} from both sides.

$$80x^{2}+8x-81x^{2}=0$$

Combine 80x^{2} and -81x^{2} to get -x^{2}.

$$-x^{2}+8x=0$$

Factor out x.

$$x\left(-x+8\right)=0$$

To find equation solutions, solve x=0 and -x+8=0.

$$x=0 x=8$$

Substitute 0 for x in the equation \sqrt{10x+1}-\sqrt{2x}=\sqrt{3x+1}.

$$\sqrt{10\times 0+1}-\sqrt{2\times 0}=\sqrt{3\times 0+1}$$

Simplify. The value x=0 satisfies the equation.

$$1=1$$

Substitute 8 for x in the equation \sqrt{10x+1}-\sqrt{2x}=\sqrt{3x+1}.

$$\sqrt{10\times 8+1}-\sqrt{2\times 8}=\sqrt{3\times 8+1}$$

Simplify. The value x=8 satisfies the equation.

$$5=5$$

List all solutions of \sqrt{10x+1}-\sqrt{2x}=\sqrt{3x+1}.

$$x=0 x=8$$