Solve (x-4)^2=225

$$( x - 4 ) ^ { 2 } = 225$$

Solve for x

$$x=19$$

$$x=-11$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16=225$$

Subtract 225 from both sides.

$$x^{2}-8x+16-225=0$$

Subtract 225 from 16 to get -209.

$$x^{2}-8x-209=0$$

To solve the equation, factor x^{2}-8x-209 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.

$$a+b=-8 ab=-209$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -209.

$$1,-209 11,-19$$

Calculate the sum for each pair.

$$1-209=-208 11-19=-8$$

The solution is the pair that gives sum -8.

$$a=-19 b=11$$

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

$$\left(x-19\right)\left(x+11\right)$$

To find equation solutions, solve x-19=0 and x+11=0.

$$x=19 x=-11$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16=225$$

Subtract 225 from both sides.

$$x^{2}-8x+16-225=0$$

Subtract 225 from 16 to get -209.

$$x^{2}-8x-209=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-209. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\left(-209\right)=-209$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -209.

$$1,-209 11,-19$$

Calculate the sum for each pair.

$$1-209=-208 11-19=-8$$

The solution is the pair that gives sum -8.

$$a=-19 b=11$$

Rewrite x^{2}-8x-209 as \left(x^{2}-19x\right)+\left(11x-209\right).

$$\left(x^{2}-19x\right)+\left(11x-209\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(x-19\right)+11\left(x-19\right)$$

Factor out common term x-19 by using distributive property.

$$\left(x-19\right)\left(x+11\right)$$

To find equation solutions, solve x-19=0 and x+11=0.

$$x=19 x=-11$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16=225$$

Subtract 225 from both sides.

$$x^{2}-8x+16-225=0$$

Subtract 225 from 16 to get -209.

$$x^{2}-8x-209=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -209 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-209\right)}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-209\right)}}{2}$$

Multiply -4 times -209.

$$x=\frac{-\left(-8\right)±\sqrt{64+836}}{2}$$

Add 64 to 836.

$$x=\frac{-\left(-8\right)±\sqrt{900}}{2}$$

Take the square root of 900.

$$x=\frac{-\left(-8\right)±30}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±30}{2}$$

Now solve the equation x=\frac{8±30}{2} when ± is plus. Add 8 to 30.

$$x=\frac{38}{2}$$

Divide 38 by 2.

$$x=19$$

Now solve the equation x=\frac{8±30}{2} when ± is minus. Subtract 30 from 8.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=19 x=-11$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{225}$$

Simplify.

$$x-4=15 x-4=-15$$

Add 4 to both sides of the equation.

$$x=19 x=-11$$