Solve sqrt{x-2}+8=x

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

Solve for x

$$x=11$$

Solution Steps

Subtract 8 from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

$$x-2=x^{2}-16x+64$$

Subtract x^{2} from both sides.

$$x-2-x^{2}=-16x+64$$

Add 16x to both sides.

$$x-2-x^{2}+16x=64$$

Combine x and 16x to get 17x.

$$17x-2-x^{2}=64$$

Subtract 64 from both sides.

$$17x-2-x^{2}-64=0$$

Subtract 64 from -2 to get -66.

$$17x-66-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+17x-66=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-66. To find a and b, set up a system to be solved.

$$a+b=17 ab=-\left(-66\right)=66$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 66.

$$1,66 2,33 3,22 6,11$$

Calculate the sum for each pair.

$$1+66=67 2+33=35 3+22=25 6+11=17$$

The solution is the pair that gives sum 17.

$$a=11 b=6$$

Rewrite -x^{2}+17x-66 as \left(-x^{2}+11x\right)+\left(6x-66\right).

$$\left(-x^{2}+11x\right)+\left(6x-66\right)$$

Factor out -x in the first and 6 in the second group.

$$-x\left(x-11\right)+6\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(-x+6\right)$$

To find equation solutions, solve x-11=0 and -x+6=0.

$$x=11 x=6$$

Substitute 11 for x in the equation \sqrt{x-2}+8=x.

$$\sqrt{11-2}+8=11$$

Simplify. The value x=11 satisfies the equation.

$$11=11$$

Substitute 6 for x in the equation \sqrt{x-2}+8=x.

$$\sqrt{6-2}+8=6$$

Simplify. The value x=6 does not satisfy the equation.

$$10=6$$

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

$$x=11$$