Solve (v+8)^2=0

$$( v + 8 ) ^ { 2 } = 0$$

Solve for v

$$v=-8$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+8\right)^{2}.

$$v^{2}+16v+64=0$$

To solve the equation, factor v^{2}+16v+64 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.

$$a+b=16 ab=64$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 64.

$$1,64 2,32 4,16 8,8$$

Calculate the sum for each pair.

$$1+64=65 2+32=34 4+16=20 8+8=16$$

The solution is the pair that gives sum 16.

$$a=8 b=8$$

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

$$\left(v+8\right)\left(v+8\right)$$

Rewrite as a binomial square.

$$\left(v+8\right)^{2}$$

To find equation solution, solve v+8=0.

$$v=-8$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+8\right)^{2}.

$$v^{2}+16v+64=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as v^{2}+av+bv+64. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\times 64=64$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 64.

$$1,64 2,32 4,16 8,8$$

Calculate the sum for each pair.

$$1+64=65 2+32=34 4+16=20 8+8=16$$

The solution is the pair that gives sum 16.

$$a=8 b=8$$

Rewrite v^{2}+16v+64 as \left(v^{2}+8v\right)+\left(8v+64\right).

$$\left(v^{2}+8v\right)+\left(8v+64\right)$$

Factor out v in the first and 8 in the second group.

$$v\left(v+8\right)+8\left(v+8\right)$$

Factor out common term v+8 by using distributive property.

$$\left(v+8\right)\left(v+8\right)$$

Rewrite as a binomial square.

$$\left(v+8\right)^{2}$$

To find equation solution, solve v+8=0.

$$v=-8$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+8\right)^{2}.

$$v^{2}+16v+64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-16±\sqrt{16^{2}-4\times 64}}{2}$$

Square 16.

$$v=\frac{-16±\sqrt{256-4\times 64}}{2}$$

Multiply -4 times 64.

$$v=\frac{-16±\sqrt{256-256}}{2}$$

Add 256 to -256.

$$v=\frac{-16±\sqrt{0}}{2}$$

Take the square root of 0.

$$v=-\frac{16}{2}$$

Divide -16 by 2.

$$v=-8$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(v+8\right)^{2}}=\sqrt{0}$$

Simplify.

$$v+8=0 v+8=0$$

Subtract 8 from both sides of the equation.

$$v=-8 v=-8$$

The equation is now solved. Solutions are the same.

$$v=-8$$