Solve sqrt{2x+2}-sqrt{x+8}=1

$$\sqrt { 2 x + 2 } - \sqrt { x + 8 } = 1$$

Solve for x

$$x=17$$

Solution Steps

Subtract -\sqrt{x+8} from both sides of the equation.

$$\sqrt{2x+2}=1+\sqrt{x+8}$$

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{x+8}\right)^{2}.

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

Calculate \sqrt{x+8} to the power of 2 and get x+8.

$$2x+2=1+2\sqrt{x+8}+x+8$$

Add 1 and 8 to get 9.

$$2x+2=9+2\sqrt{x+8}+x$$

Subtract 9+x from both sides of the equation.

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

To find the opposite of 9+x, find the opposite of each term.

$$2x+2-9-x=2\sqrt{x+8}$$

Subtract 9 from 2 to get -7.

$$2x-7-x=2\sqrt{x+8}$$

Combine 2x and -x to get x.

$$x-7=2\sqrt{x+8}$$

Square both sides of the equation.

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

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

$$x^{2}-14x+49=\left(2\sqrt{x+8}\right)^{2}$$

Expand \left(2\sqrt{x+8}\right)^{2}.

$$x^{2}-14x+49=2^{2}\left(\sqrt{x+8}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}-14x+49=4\left(\sqrt{x+8}\right)^{2}$$

Calculate \sqrt{x+8} to the power of 2 and get x+8.

$$x^{2}-14x+49=4\left(x+8\right)$$

Use the distributive property to multiply 4 by x+8.

$$x^{2}-14x+49=4x+32$$

Subtract 4x from both sides.

$$x^{2}-14x+49-4x=32$$

Combine -14x and -4x to get -18x.

$$x^{2}-18x+49=32$$

Subtract 32 from both sides.

$$x^{2}-18x+49-32=0$$

Subtract 32 from 49 to get 17.

$$x^{2}-18x+17=0$$

To solve the equation, factor x^{2}-18x+17 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-18 ab=17$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-17 b=-1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-17\right)\left(x-1\right)$$

To find equation solutions, solve x-17=0 and x-1=0.

$$x=17 x=1$$

Substitute 17 for x in the equation \sqrt{2x+2}-\sqrt{x+8}=1.

$$\sqrt{2\times 17+2}-\sqrt{17+8}=1$$

Simplify. The value x=17 satisfies the equation.

$$1=1$$

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

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

Simplify. The value x=1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Substitute 17 for x in the equation \sqrt{2x+2}-\sqrt{x+8}=1.

$$\sqrt{2\times 17+2}-\sqrt{17+8}=1$$

Simplify. The value x=17 satisfies the equation.

$$1=1$$

Equation \sqrt{2x+2}=\sqrt{x+8}+1 has a unique solution.

$$x=17$$