Solve (v+10)^2=121

$$( v + 10 ) ^ { 2 } = 121$$

Solve for v

$$v=1$$

$$v=-21$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+10\right)^{2}.

$$v^{2}+20v+100=121$$

Subtract 121 from both sides.

$$v^{2}+20v+100-121=0$$

Subtract 121 from 100 to get -21.

$$v^{2}+20v-21=0$$

To solve the equation, factor v^{2}+20v-21 using formula v^{2}+\left(a+b\right)v+ab=\left(v+a\right)\left(v+b\right). To find a and b, set up a system to be solved.

$$a+b=20 ab=-21$$

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 -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 20.

$$a=-1 b=21$$

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

$$\left(v-1\right)\left(v+21\right)$$

To find equation solutions, solve v-1=0 and v+21=0.

$$v=1 v=-21$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+10\right)^{2}.

$$v^{2}+20v+100=121$$

Subtract 121 from both sides.

$$v^{2}+20v+100-121=0$$

Subtract 121 from 100 to get -21.

$$v^{2}+20v-21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as v^{2}+av+bv-21. To find a and b, set up a system to be solved.

$$a+b=20 ab=1\left(-21\right)=-21$$

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 -21.

$$-1,21 -3,7$$

Calculate the sum for each pair.

$$-1+21=20 -3+7=4$$

The solution is the pair that gives sum 20.

$$a=-1 b=21$$

Rewrite v^{2}+20v-21 as \left(v^{2}-v\right)+\left(21v-21\right).

$$\left(v^{2}-v\right)+\left(21v-21\right)$$

Factor out v in the first and 21 in the second group.

$$v\left(v-1\right)+21\left(v-1\right)$$

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

$$\left(v-1\right)\left(v+21\right)$$

To find equation solutions, solve v-1=0 and v+21=0.

$$v=1 v=-21$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+10\right)^{2}.

$$v^{2}+20v+100=121$$

Subtract 121 from both sides.

$$v^{2}+20v+100-121=0$$

Subtract 121 from 100 to get -21.

$$v^{2}+20v-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-20±\sqrt{20^{2}-4\left(-21\right)}}{2}$$

Square 20.

$$v=\frac{-20±\sqrt{400-4\left(-21\right)}}{2}$$

Multiply -4 times -21.

$$v=\frac{-20±\sqrt{400+84}}{2}$$

Add 400 to 84.

$$v=\frac{-20±\sqrt{484}}{2}$$

Take the square root of 484.

$$v=\frac{-20±22}{2}$$

Now solve the equation v=\frac{-20±22}{2} when ± is plus. Add -20 to 22.

$$v=\frac{2}{2}$$

Divide 2 by 2.

$$v=1$$

Now solve the equation v=\frac{-20±22}{2} when ± is minus. Subtract 22 from -20.

$$v=-\frac{42}{2}$$

Divide -42 by 2.

$$v=-21$$

The equation is now solved.

$$v=1 v=-21$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(v+10\right)^{2}}=\sqrt{121}$$

Simplify.

$$v+10=11 v+10=-11$$

Subtract 10 from both sides of the equation.

$$v=1 v=-21$$