Solve sqrt{x-6}=8-x

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

Solve for x

$$x=7$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Subtract 64 from both sides.

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

Subtract 64 from -6 to get -70.

$$x-70=-16x+x^{2}$$

Add 16x to both sides.

$$x-70+16x=x^{2}$$

Combine x and 16x to get 17x.

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

Subtract x^{2} from both sides.

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

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 because the left and the right hand side have opposite signs.

$$2=-2$$

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.

$$1=1$$

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

$$x=7$$