Solve (v+4)^2=2v^2+2v+9

$$( v + 4 ) ^ { 2 } = 2 v ^ { 2 } + 2 v + 9$$

Solve for v

$$v=-1$$

$$v=7$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+4\right)^{2}.

$$v^{2}+8v+16=2v^{2}+2v+9$$

Subtract 2v^{2} from both sides.

$$v^{2}+8v+16-2v^{2}=2v+9$$

Combine v^{2} and -2v^{2} to get -v^{2}.

$$-v^{2}+8v+16=2v+9$$

Subtract 2v from both sides.

$$-v^{2}+8v+16-2v=9$$

Combine 8v and -2v to get 6v.

$$-v^{2}+6v+16=9$$

Subtract 9 from both sides.

$$-v^{2}+6v+16-9=0$$

Subtract 9 from 16 to get 7.

$$-v^{2}+6v+7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -v^{2}+av+bv+7. To find a and b, set up a system to be solved.

$$a+b=6 ab=-7=-7$$

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.

$$a=7 b=-1$$

Rewrite -v^{2}+6v+7 as \left(-v^{2}+7v\right)+\left(-v+7\right).

$$\left(-v^{2}+7v\right)+\left(-v+7\right)$$

Factor out -v in the first and -1 in the second group.

$$-v\left(v-7\right)-\left(v-7\right)$$

Factor out common term v-7 by using distributive property.

$$\left(v-7\right)\left(-v-1\right)$$

To find equation solutions, solve v-7=0 and -v-1=0.

$$v=7 v=-1$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+4\right)^{2}.

$$v^{2}+8v+16=2v^{2}+2v+9$$

Subtract 2v^{2} from both sides.

$$v^{2}+8v+16-2v^{2}=2v+9$$

Combine v^{2} and -2v^{2} to get -v^{2}.

$$-v^{2}+8v+16=2v+9$$

Subtract 2v from both sides.

$$-v^{2}+8v+16-2v=9$$

Combine 8v and -2v to get 6v.

$$-v^{2}+6v+16=9$$

Subtract 9 from both sides.

$$-v^{2}+6v+16-9=0$$

Subtract 9 from 16 to get 7.

$$-v^{2}+6v+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Square 6.

$$v=\frac{-6±\sqrt{36-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$v=\frac{-6±\sqrt{36+4\times 7}}{2\left(-1\right)}$$

Multiply 4 times 7.

$$v=\frac{-6±\sqrt{36+28}}{2\left(-1\right)}$$

Add 36 to 28.

$$v=\frac{-6±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$v=\frac{-6±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$v=\frac{-6±8}{-2}$$

Now solve the equation v=\frac{-6±8}{-2} when ± is plus. Add -6 to 8.

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

Divide 2 by -2.

$$v=-1$$

Now solve the equation v=\frac{-6±8}{-2} when ± is minus. Subtract 8 from -6.

$$v=-\frac{14}{-2}$$

Divide -14 by -2.

$$v=7$$

The equation is now solved.

$$v=-1 v=7$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(v+4\right)^{2}.

$$v^{2}+8v+16=2v^{2}+2v+9$$

Subtract 2v^{2} from both sides.

$$v^{2}+8v+16-2v^{2}=2v+9$$

Combine v^{2} and -2v^{2} to get -v^{2}.

$$-v^{2}+8v+16=2v+9$$

Subtract 2v from both sides.

$$-v^{2}+8v+16-2v=9$$

Combine 8v and -2v to get 6v.

$$-v^{2}+6v+16=9$$

Subtract 16 from both sides.

$$-v^{2}+6v=9-16$$

Subtract 16 from 9 to get -7.

$$-v^{2}+6v=-7$$

Divide both sides by -1.

$$\frac{-v^{2}+6v}{-1}=-\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$v^{2}+\frac{6}{-1}v=-\frac{7}{-1}$$

Divide 6 by -1.

$$v^{2}-6v=-\frac{7}{-1}$$

Divide -7 by -1.

$$v^{2}-6v=7$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-6v+\left(-3\right)^{2}=7+\left(-3\right)^{2}$$

Square -3.

$$v^{2}-6v+9=7+9$$

Add 7 to 9.

$$v^{2}-6v+9=16$$

Factor v^{2}-6v+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-3\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-3\right)^{2}}=\sqrt{16}$$

Simplify.

$$v-3=4 v-3=-4$$

Add 3 to both sides of the equation.

$$v=7 v=-1$$