Solve (x-9)^2=64

$$( x - 9 ) ^ { 2 } = 64$$

Solve for x

$$x=17$$

$$x=1$$

Steps Using Factoring

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

$$x^{2}-18x+81=64$$

Subtract 64 from both sides.

$$x^{2}-18x+81-64=0$$

Subtract 64 from 81 to get 17.

$$x^{2}-18x+17=0$$

To solve the equation, factor x^{2}-18x+17 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.

$$a+b=-18 ab=17$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-17 b=-1$$

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

$$\left(x-17\right)\left(x-1\right)$$

To find equation solutions, solve x-17=0 and x-1=0.

$$x=17 x=1$$

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}.

$$x^{2}-18x+81=64$$

Subtract 64 from both sides.

$$x^{2}-18x+81-64=0$$

Subtract 64 from 81 to get 17.

$$x^{2}-18x+17=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+17. To find a and b, set up a system to be solved.

$$a+b=-18 ab=1\times 17=17$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-17 b=-1$$

Rewrite x^{2}-18x+17 as \left(x^{2}-17x\right)+\left(-x+17\right).

$$\left(x^{2}-17x\right)+\left(-x+17\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-17\right)-\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(x-1\right)$$

To find equation solutions, solve x-17=0 and x-1=0.

$$x=17 x=1$$

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}.

$$x^{2}-18x+81=64$$

Subtract 64 from both sides.

$$x^{2}-18x+81-64=0$$

Subtract 64 from 81 to get 17.

$$x^{2}-18x+17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 17}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 17}}{2}$$

Multiply -4 times 17.

$$x=\frac{-\left(-18\right)±\sqrt{324-68}}{2}$$

Add 324 to -68.

$$x=\frac{-\left(-18\right)±\sqrt{256}}{2}$$

Take the square root of 256.

$$x=\frac{-\left(-18\right)±16}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±16}{2}$$

Now solve the equation x=\frac{18±16}{2} when ± is plus. Add 18 to 16.

$$x=\frac{34}{2}$$

Divide 34 by 2.

$$x=17$$

Now solve the equation x=\frac{18±16}{2} when ± is minus. Subtract 16 from 18.

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=17 x=1$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{64}$$

Simplify.

$$x-9=8 x-9=-8$$

Add 9 to both sides of the equation.

$$x=17 x=1$$