Solve sqrt{9x+4}=sqrt{7x+1}+1

$$\sqrt { 9 x + 4 } = \sqrt { 7 x + 1 } + 1$$

Solve for x

$$x=5$$

$$x=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{9x+4}\right)^{2}=\left(\sqrt{7x+1}+1\right)^{2}$$

Calculate \sqrt{9x+4} to the power of 2 and get 9x+4.

$$9x+4=\left(\sqrt{7x+1}+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{7x+1}+1\right)^{2}.

$$9x+4=\left(\sqrt{7x+1}\right)^{2}+2\sqrt{7x+1}+1$$

Calculate \sqrt{7x+1} to the power of 2 and get 7x+1.

$$9x+4=7x+1+2\sqrt{7x+1}+1$$

Add 1 and 1 to get 2.

$$9x+4=7x+2+2\sqrt{7x+1}$$

Subtract 7x+2 from both sides of the equation.

$$9x+4-\left(7x+2\right)=2\sqrt{7x+1}$$

To find the opposite of 7x+2, find the opposite of each term.

$$9x+4-7x-2=2\sqrt{7x+1}$$

Combine 9x and -7x to get 2x.

$$2x+4-2=2\sqrt{7x+1}$$

Subtract 2 from 4 to get 2.

$$2x+2=2\sqrt{7x+1}$$

Square both sides of the equation.

$$\left(2x+2\right)^{2}=\left(2\sqrt{7x+1}\right)^{2}$$

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

$$4x^{2}+8x+4=\left(2\sqrt{7x+1}\right)^{2}$$

Expand \left(2\sqrt{7x+1}\right)^{2}.

$$4x^{2}+8x+4=2^{2}\left(\sqrt{7x+1}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}+8x+4=4\left(\sqrt{7x+1}\right)^{2}$$

Calculate \sqrt{7x+1} to the power of 2 and get 7x+1.

$$4x^{2}+8x+4=4\left(7x+1\right)$$

Use the distributive property to multiply 4 by 7x+1.

$$4x^{2}+8x+4=28x+4$$

Subtract 28x from both sides.

$$4x^{2}+8x+4-28x=4$$

Combine 8x and -28x to get -20x.

$$4x^{2}-20x+4=4$$

Subtract 4 from both sides.

$$4x^{2}-20x+4-4=0$$

Subtract 4 from 4 to get 0.

$$4x^{2}-20x=0$$

Factor out x.

$$x\left(4x-20\right)=0$$

To find equation solutions, solve x=0 and 4x-20=0.

$$x=0 x=5$$

Substitute 0 for x in the equation \sqrt{9x+4}=\sqrt{7x+1}+1.

$$\sqrt{9\times 0+4}=\sqrt{7\times 0+1}+1$$

Simplify. The value x=0 satisfies the equation.

$$2=2$$

Substitute 5 for x in the equation \sqrt{9x+4}=\sqrt{7x+1}+1.

$$\sqrt{9\times 5+4}=\sqrt{7\times 5+1}+1$$

Simplify. The value x=5 satisfies the equation.

$$7=7$$

List all solutions of \sqrt{9x+4}=\sqrt{7x+1}+1.

$$x=0 x=5$$