Solve sqrt{x-6}-8=-x

$$\sqrt { x - 6 } - 8 = - x$$

Solve for x

$$x=7$$

Solution Steps

Subtract -8 from both sides of the equation.

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

Square both sides of the equation.

$$\left(\sqrt{x-6}\right)^{2}=\left(-x+8\right)^{2}$$

Calculate \sqrt{x-6} to the power of 2 and get x-6.

$$x-6=\left(-x+8\right)^{2}$$

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

$$x-6=\left(-x\right)^{2}+16\left(-x\right)+64$$

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

$$x-6=x^{2}+16\left(-x\right)+64$$

Subtract x^{2} from both sides.

$$x-6-x^{2}=16\left(-x\right)+64$$

Subtract 16\left(-x\right) from both sides.

$$x-6-x^{2}-16\left(-x\right)=64$$

Subtract 64 from both sides.

$$x-6-x^{2}-16\left(-x\right)-64=0$$

Multiply -1 and 16 to get -16.

$$x-6-x^{2}-16\left(-1\right)x-64=0$$

Multiply -16 and -1 to get 16.

$$x-6-x^{2}+16x-64=0$$

Combine x and 16x to get 17x.

$$17x-6-x^{2}-64=0$$

Subtract 64 from -6 to get -70.

$$17x-70-x^{2}=0$$

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

$$-x^{2}+17x-70=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-70. To find a and b, set up a system to be solved.

$$a+b=17 ab=-\left(-70\right)=70$$

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 70.

$$1,70 2,35 5,14 7,10$$

Calculate the sum for each pair.

$$1+70=71 2+35=37 5+14=19 7+10=17$$

The solution is the pair that gives sum 17.

$$a=10 b=7$$

Rewrite -x^{2}+17x-70 as \left(-x^{2}+10x\right)+\left(7x-70\right).

$$\left(-x^{2}+10x\right)+\left(7x-70\right)$$

Factor out -x in the first and 7 in the second group.

$$-x\left(x-10\right)+7\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(-x+7\right)$$

To find equation solutions, solve x-10=0 and -x+7=0.

$$x=10 x=7$$

Substitute 10 for x in the equation \sqrt{x-6}-8=-x.

$$\sqrt{10-6}-8=-10$$

Simplify. The value x=10 does not satisfy the equation.

$$-6=-10$$

Substitute 7 for x in the equation \sqrt{x-6}-8=-x.

$$\sqrt{7-6}-8=-7$$

Simplify. The value x=7 satisfies the equation.

$$-7=-7$$

Equation \sqrt{x-6}=8-x has a unique solution.

$$x=7$$