Solve sqrt{x-6}=x-6

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

Solve for x

$$x=6$$

$$x=7$$

Solution Steps

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x-6=x^{2}-12x+36$$

Subtract x^{2} from both sides.

$$x-6-x^{2}=-12x+36$$

Add 12x to both sides.

$$x-6-x^{2}+12x=36$$

Combine x and 12x to get 13x.

$$13x-6-x^{2}=36$$

Subtract 36 from both sides.

$$13x-6-x^{2}-36=0$$

Subtract 36 from -6 to get -42.

$$13x-42-x^{2}=0$$

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

$$-x^{2}+13x-42=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-42. To find a and b, set up a system to be solved.

$$a+b=13 ab=-\left(-42\right)=42$$

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 42.

$$1,42 2,21 3,14 6,7$$

Calculate the sum for each pair.

$$1+42=43 2+21=23 3+14=17 6+7=13$$

The solution is the pair that gives sum 13.

$$a=7 b=6$$

Rewrite -x^{2}+13x-42 as \left(-x^{2}+7x\right)+\left(6x-42\right).

$$\left(-x^{2}+7x\right)+\left(6x-42\right)$$

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

$$-x\left(x-7\right)+6\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x+6\right)$$

To find equation solutions, solve x-7=0 and -x+6=0.

$$x=7 x=6$$

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

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

Simplify. The value x=7 satisfies the equation.

$$1=1$$

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

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

Simplify. The value x=6 satisfies the equation.

$$0=0$$

List all solutions of \sqrt{x-6}=x-6.

$$x=7 x=6$$