Solve (x-9)^2=25

$$( x - 9 ) ^ { 2 } = 25$$

Solve for x

$$x=14$$

$$x=4$$

Steps Using Factoring

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

$$x^{2}-18x+81=25$$

Subtract 25 from both sides.

$$x^{2}-18x+81-25=0$$

Subtract 25 from 81 to get 56.

$$x^{2}-18x+56=0$$

To solve the equation, factor x^{2}-18x+56 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.

$$a+b=-18 ab=56$$

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 56.

$$-1,-56 -2,-28 -4,-14 -7,-8$$

Calculate the sum for each pair.

$$-1-56=-57 -2-28=-30 -4-14=-18 -7-8=-15$$

The solution is the pair that gives sum -18.

$$a=-14 b=-4$$

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

$$\left(x-14\right)\left(x-4\right)$$

To find equation solutions, solve x-14=0 and x-4=0.

$$x=14 x=4$$

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}.

$$x^{2}-18x+81=25$$

Subtract 25 from both sides.

$$x^{2}-18x+81-25=0$$

Subtract 25 from 81 to get 56.

$$x^{2}-18x+56=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+56. To find a and b, set up a system to be solved.

$$a+b=-18 ab=1\times 56=56$$

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 56.

$$-1,-56 -2,-28 -4,-14 -7,-8$$

Calculate the sum for each pair.

$$-1-56=-57 -2-28=-30 -4-14=-18 -7-8=-15$$

The solution is the pair that gives sum -18.

$$a=-14 b=-4$$

Rewrite x^{2}-18x+56 as \left(x^{2}-14x\right)+\left(-4x+56\right).

$$\left(x^{2}-14x\right)+\left(-4x+56\right)$$

Factor out x in the first and -4 in the second group.

$$x\left(x-14\right)-4\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(x-4\right)$$

To find equation solutions, solve x-14=0 and x-4=0.

$$x=14 x=4$$

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}.

$$x^{2}-18x+81=25$$

Subtract 25 from both sides.

$$x^{2}-18x+81-25=0$$

Subtract 25 from 81 to get 56.

$$x^{2}-18x+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 56}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 56}}{2}$$

Multiply -4 times 56.

$$x=\frac{-\left(-18\right)±\sqrt{324-224}}{2}$$

Add 324 to -224.

$$x=\frac{-\left(-18\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-\left(-18\right)±10}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±10}{2}$$

Now solve the equation x=\frac{18±10}{2} when ± is plus. Add 18 to 10.

$$x=\frac{28}{2}$$

Divide 28 by 2.

$$x=14$$

Now solve the equation x=\frac{18±10}{2} when ± is minus. Subtract 10 from 18.

$$x=\frac{8}{2}$$

Divide 8 by 2.

$$x=4$$

The equation is now solved.

$$x=14 x=4$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-9=5 x-9=-5$$

Add 9 to both sides of the equation.

$$x=14 x=4$$