Solve (x+9)^2=100=

$$( x + 9 ) ^ { 2 } = 100 =$$

Solve for x

$$x=1$$

$$x=-19$$

Steps Using Factoring

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

$$x^{2}+18x+81=100$$

Subtract 100 from both sides.

$$x^{2}+18x+81-100=0$$

Subtract 100 from 81 to get -19.

$$x^{2}+18x-19=0$$

To solve the equation, factor x^{2}+18x-19 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.

$$a+b=18 ab=-19$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=19$$

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

$$\left(x-1\right)\left(x+19\right)$$

To find equation solutions, solve x-1=0 and x+19=0.

$$x=1 x=-19$$

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}.

$$x^{2}+18x+81=100$$

Subtract 100 from both sides.

$$x^{2}+18x+81-100=0$$

Subtract 100 from 81 to get -19.

$$x^{2}+18x-19=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-19. To find a and b, set up a system to be solved.

$$a+b=18 ab=1\left(-19\right)=-19$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=-1 b=19$$

Rewrite x^{2}+18x-19 as \left(x^{2}-x\right)+\left(19x-19\right).

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

Factor out x in the first and 19 in the second group.

$$x\left(x-1\right)+19\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+19\right)$$

To find equation solutions, solve x-1=0 and x+19=0.

$$x=1 x=-19$$

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}.

$$x^{2}+18x+81=100$$

Subtract 100 from both sides.

$$x^{2}+18x+81-100=0$$

Subtract 100 from 81 to get -19.

$$x^{2}+18x-19=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and -19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-19\right)}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-19\right)}}{2}$$

Multiply -4 times -19.

$$x=\frac{-18±\sqrt{324+76}}{2}$$

Add 324 to 76.

$$x=\frac{-18±\sqrt{400}}{2}$$

Take the square root of 400.

$$x=\frac{-18±20}{2}$$

Now solve the equation x=\frac{-18±20}{2} when ± is plus. Add -18 to 20.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-18±20}{2} when ± is minus. Subtract 20 from -18.

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

Divide -38 by 2.

$$x=-19$$

The equation is now solved.

$$x=1 x=-19$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{100}$$

Simplify.

$$x+9=10 x+9=-10$$

Subtract 9 from both sides of the equation.

$$x=1 x=-19$$