Algebra Examples

$9 = v + \frac{4}{v} + 12$

Step 1

Rewrite the equation as $v + \frac{4}{v} + 12 = 9$ .

$v + \frac{4}{v} + 12 = 9$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, v, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$v$

Step 3

Multiply each term in $v + \frac{4}{v} + 12 = 9$ by $v$ to eliminate the fractions.

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Step 3.1

Multiply each term in $v + \frac{4}{v} + 12 = 9$ by $v$ .

$v \cdot v + \frac{4}{v} v + 12 v = 9 v$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $v$ by $v$ .

$v^{2} + \frac{4}{v} v + 12 v = 9 v$

Step 3.2.1.2

Cancel the common factor of $v$ .

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Step 3.2.1.2.1

Cancel the common factor.

$v^{2} + \frac{4}{v} v + 12 v = 9 v$

Step 3.2.1.2.2

Rewrite the expression.

$v^{2} + 4 + 12 v = 9 v$

Step 4

Solve the equation.

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Step 4.1

Move all terms containing $v$ to the left side of the equation.

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Step 4.1.1

Subtract $9 v$ from both sides of the equation.

$v^{2} + 4 + 12 v - 9 v = 0$

Step 4.1.2

Subtract $9 v$ from $12 v$ .

$v^{2} + 4 + 3 v = 0$

Step 4.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.3

Substitute the values $a = 1$ , $b = 3$ , and $c = 4$ into the quadratic formula and solve for $v$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Raise $3$ to the power of $2$ .

$v = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

$v = \frac{- 3 \pm \sqrt{9 - 4 \cdot 4}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $- 4$ by $4$ .

$v = \frac{- 3 \pm \sqrt{9 - 16}}{2 \cdot 1}$

Step 4.4.1.3

Subtract $16$ from $9$ .

$v = \frac{- 3 \pm \sqrt{- 7}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$v = \frac{- 3 \pm \sqrt{- 1 \cdot 7}}{2 \cdot 1}$

Step 4.4.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$v = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot 1}$

Step 4.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$v = \frac{- 3 \pm i \sqrt{7}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$v = \frac{- 3 \pm i \sqrt{7}}{2}$

Step 4.5

The final answer is the combination of both solutions.

$v = - \frac{3 - i \sqrt{7}}{2}, - \frac{3 + i \sqrt{7}}{2}$