Solve (x-9)^2=9

$$( x - 9 ) ^ { 2 } = 9$$

Solve for x

$$x=12$$

$$x=6$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-9\right)^{2}.

$$x^{2}-18x+81=9$$

Subtract 9 from both sides.

$$x^{2}-18x+81-9=0$$

Subtract 9 from 81 to get 72.

$$x^{2}-18x+72=0$$

To solve the equation, factor x^{2}-18x+72 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.

$$a+b=-18 ab=72$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -18.

$$a=-12 b=-6$$

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

$$\left(x-12\right)\left(x-6\right)$$

To find equation solutions, solve x-12=0 and x-6=0.

$$x=12 x=6$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-9\right)^{2}.

$$x^{2}-18x+81=9$$

Subtract 9 from both sides.

$$x^{2}-18x+81-9=0$$

Subtract 9 from 81 to get 72.

$$x^{2}-18x+72=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+72. To find a and b, set up a system to be solved.

$$a+b=-18 ab=1\times 72=72$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -18.

$$a=-12 b=-6$$

Rewrite x^{2}-18x+72 as \left(x^{2}-12x\right)+\left(-6x+72\right).

$$\left(x^{2}-12x\right)+\left(-6x+72\right)$$

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

$$x\left(x-12\right)-6\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x-6\right)$$

To find equation solutions, solve x-12=0 and x-6=0.

$$x=12 x=6$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-9\right)^{2}.

$$x^{2}-18x+81=9$$

Subtract 9 from both sides.

$$x^{2}-18x+81-9=0$$

Subtract 9 from 81 to get 72.

$$x^{2}-18x+72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 72}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 72}}{2}$$

Multiply -4 times 72.

$$x=\frac{-\left(-18\right)±\sqrt{324-288}}{2}$$

Add 324 to -288.

$$x=\frac{-\left(-18\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-\left(-18\right)±6}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±6}{2}$$

Now solve the equation x=\frac{18±6}{2} when ± is plus. Add 18 to 6.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{18±6}{2} when ± is minus. Subtract 6 from 18.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

The equation is now solved.

$$x=12 x=6$$

Steps for Completing the Square

Take the square root of both sides of the equation.

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

Simplify.

$$x-9=3 x-9=-3$$

Add 9 to both sides of the equation.

$$x=12 x=6$$