Solve left(x+7right)^2=81

$${ \left(x+7 \right) }^{ 2 } = 81$$

Solve for x

$$x=2$$

$$x=-16$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=81$$

Subtract 81 from both sides.

$$x^{2}+14x+49-81=0$$

Subtract 81 from 49 to get -32.

$$x^{2}+14x-32=0$$

To solve the equation, factor x^{2}+14x-32 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.

$$a+b=14 ab=-32$$

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. List all such integer pairs that give product -32.

$$-1,32 -2,16 -4,8$$

Calculate the sum for each pair.

$$-1+32=31 -2+16=14 -4+8=4$$

The solution is the pair that gives sum 14.

$$a=-2 b=16$$

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

$$\left(x-2\right)\left(x+16\right)$$

To find equation solutions, solve x-2=0 and x+16=0.

$$x=2 x=-16$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=81$$

Subtract 81 from both sides.

$$x^{2}+14x+49-81=0$$

Subtract 81 from 49 to get -32.

$$x^{2}+14x-32=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-32. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\left(-32\right)=-32$$

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. List all such integer pairs that give product -32.

$$-1,32 -2,16 -4,8$$

Calculate the sum for each pair.

$$-1+32=31 -2+16=14 -4+8=4$$

The solution is the pair that gives sum 14.

$$a=-2 b=16$$

Rewrite x^{2}+14x-32 as \left(x^{2}-2x\right)+\left(16x-32\right).

$$\left(x^{2}-2x\right)+\left(16x-32\right)$$

Factor out x in the first and 16 in the second group.

$$x\left(x-2\right)+16\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x+16\right)$$

To find equation solutions, solve x-2=0 and x+16=0.

$$x=2 x=-16$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+14x+49=81$$

Subtract 81 from both sides.

$$x^{2}+14x+49-81=0$$

Subtract 81 from 49 to get -32.

$$x^{2}+14x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-32\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-32\right)}}{2}$$

Multiply -4 times -32.

$$x=\frac{-14±\sqrt{196+128}}{2}$$

Add 196 to 128.

$$x=\frac{-14±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-14±18}{2}$$

Now solve the equation x=\frac{-14±18}{2} when ± is plus. Add -14 to 18.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-14±18}{2} when ± is minus. Subtract 18 from -14.

$$x=-\frac{32}{2}$$

Divide -32 by 2.

$$x=-16$$

The equation is now solved.

$$x=2 x=-16$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{81}$$

Simplify.

$$x+7=9 x+7=-9$$

Subtract 7 from both sides of the equation.

$$x=2 x=-16$$