Solve by Completing the Square 4v^2+16v=65

$4 v^{2} + 16 v = 65$

Step 1

Divide each term in $4 v^{2} + 16 v = 65$ by $4$ and simplify.

Divide each term in $4 v^{2} + 16 v = 65$ by $4$ .

$\frac{4 v^{2}}{4} + \frac{16 v}{4} = \frac{65}{4}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 v^{2}}{4} + \frac{16 v}{4} = \frac{65}{4}$

Divide $v^{2}$ by $1$ .

$v^{2} + \frac{16 v}{4} = \frac{65}{4}$

Cancel the common factor of $16$ and $4$ .

Factor $4$ out of $16 v$ .

$v^{2} + \frac{4 (4 v)}{4} = \frac{65}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$v^{2} + \frac{4 (4 v)}{4 (1)} = \frac{65}{4}$

Cancel the common factor.

$v^{2} + \frac{4 (4 v)}{4 \cdot 1} = \frac{65}{4}$

Rewrite the expression.

$v^{2} + \frac{4 v}{1} = \frac{65}{4}$

Divide $4 v$ by $1$ .

$v^{2} + 4 v = \frac{65}{4}$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(2)}^{2}$

Step 3

Add the term to each side of the equation.

$v^{2} + 4 v + {(2)}^{2} = \frac{65}{4} + {(2)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $2$ to the power of $2$ .

$v^{2} + 4 v + 4 = \frac{65}{4} + {(2)}^{2}$

Simplify the right side.

Simplify $\frac{65}{4} + {(2)}^{2}$ .

Raise $2$ to the power of $2$ .

$v^{2} + 4 v + 4 = \frac{65}{4} + 4$

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$v^{2} + 4 v + 4 = \frac{65}{4} + 4 \cdot \frac{4}{4}$

Combine $4$ and $\frac{4}{4}$ .

$v^{2} + 4 v + 4 = \frac{65}{4} + \frac{4 \cdot 4}{4}$

Combine the numerators over the common denominator.

$v^{2} + 4 v + 4 = \frac{65 + 4 \cdot 4}{4}$

Simplify the numerator.

Multiply $4$ by $4$ .

$v^{2} + 4 v + 4 = \frac{65 + 16}{4}$

Add $65$ and $16$ .

$v^{2} + 4 v + 4 = \frac{81}{4}$

Step 5

Factor the perfect trinomial square into ${(v + 2)}^{2}$ .

${(v + 2)}^{2} = \frac{81}{4}$

Step 6

Solve the equation for $v$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$v + 2 = \pm \sqrt{\frac{81}{4}}$

Simplify $\pm \sqrt{\frac{81}{4}}$ .

Rewrite $\sqrt{\frac{81}{4}}$ as $\frac{\sqrt{81}}{\sqrt{4}}$ .

$v + 2 = \pm \frac{\sqrt{81}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $81$ as $9^{2}$ .

$v + 2 = \pm \frac{\sqrt{9^{2}}}{\sqrt{4}}$

Pull terms out from under the radical, assuming positive real numbers.

$v + 2 = \pm \frac{9}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$v + 2 = \pm \frac{9}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$v + 2 = \pm \frac{9}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$v + 2 = \frac{9}{2}$

Move all terms not containing $v$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$v = \frac{9}{2} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$v = \frac{9}{2} - 2 \cdot \frac{2}{2}$

Combine $- 2$ and $\frac{2}{2}$ .

$v = \frac{9}{2} + \frac{- 2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$v = \frac{9 - 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$v = \frac{9 - 4}{2}$

Subtract $4$ from $9$ .

$v = \frac{5}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$v + 2 = - \frac{9}{2}$

Move all terms not containing $v$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$v = - \frac{9}{2} - 2$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$v = - \frac{9}{2} - 2 \cdot \frac{2}{2}$

Combine $- 2$ and $\frac{2}{2}$ .

$v = - \frac{9}{2} + \frac{- 2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$v = \frac{- 9 - 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$v = \frac{- 9 - 4}{2}$

Subtract $4$ from $- 9$ .

$v = \frac{- 13}{2}$

Move the negative in front of the fraction.

$v = - \frac{13}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$v = \frac{5}{2}, - \frac{13}{2}$

Solve by Completing the Square 4v^2+16v=65

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