Trigonometry Examples

$5 y + 12 = - \frac{8}{y}$

Step 1

Subtract $12$ from both sides of the equation.

$5 y = - \frac{8}{y} - 12$

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.

$1, y, 1$

Step 2.2

The LCM of one and any expression is the expression.

$y$

Step 3

Multiply each term in $5 y = - \frac{8}{y} - 12$ by $y$ to eliminate the fractions.

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Step 3.1

Multiply each term in $5 y = - \frac{8}{y} - 12$ by $y$ .

$5 y \cdot y = - \frac{8}{y} y - 12 y$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.1.1

Move $y$ .

$5 (y \cdot y) = - \frac{8}{y} y - 12 y$

Step 3.2.1.2

Multiply $y$ by $y$ .

$5 y^{2} = - \frac{8}{y} y - 12 y$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $y$ .

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Step 3.3.1.1

Move the leading negative in $- \frac{8}{y}$ into the numerator.

$5 y^{2} = \frac{- 8}{y} y - 12 y$

Step 3.3.1.2

Cancel the common factor.

$5 y^{2} = \frac{- 8}{y} y - 12 y$

Step 3.3.1.3

Rewrite the expression.

$5 y^{2} = - 8 - 12 y$

Step 4

Solve the equation.

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Step 4.1

Add $12 y$ to both sides of the equation.

$5 y^{2} + 12 y = - 8$

Step 4.2

Add $8$ to both sides of the equation.

$5 y^{2} + 12 y + 8 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = 5$ , $b = 12$ , and $c = 8$ into the quadratic formula and solve for $y$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (5 \cdot 8)}}{2 \cdot 5}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 5 \cdot 8}}{2 \cdot 5}$

Step 4.5.1.2

Multiply $- 4 \cdot 5 \cdot 8$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{- 12 \pm \sqrt{144 - 20 \cdot 8}}{2 \cdot 5}$

Step 4.5.1.2.2

Multiply $- 20$ by $8$ .

$y = \frac{- 12 \pm \sqrt{144 - 160}}{2 \cdot 5}$

Step 4.5.1.3

Subtract $160$ from $144$ .

$y = \frac{- 12 \pm \sqrt{- 16}}{2 \cdot 5}$

Step 4.5.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$y = \frac{- 12 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 5}$

Step 4.5.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$y = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 5}$

Step 4.5.1.6

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

$y = \frac{- 12 \pm i \cdot \sqrt{16}}{2 \cdot 5}$

Step 4.5.1.7

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 12 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 5}$

Step 4.5.1.8

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

$y = \frac{- 12 \pm i \cdot 4}{2 \cdot 5}$

Step 4.5.1.9

Move $4$ to the left of $i$ .

$y = \frac{- 12 \pm 4 i}{2 \cdot 5}$

Step 4.5.2

Multiply $2$ by $5$ .

$y = \frac{- 12 \pm 4 i}{10}$

Step 4.5.3

Simplify $\frac{- 12 \pm 4 i}{10}$ .

$y = \frac{- 6 \pm 2 i}{5}$

Step 4.6

The final answer is the combination of both solutions.

$y = - \frac{6}{5} + \frac{2 i}{5}, - \frac{6}{5} - \frac{2 i}{5}$