Trigonometry Examples

$y = (x + 9) + (\frac{74}{x + 5})$

Step 1

Rewrite the equation as $x + 9 + \frac{74}{x + 5} = y$ .

$x + 9 + \frac{74}{x + 5} = y$

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.

$1, 1, x + 5, 1$

Step 2.2

Remove parentheses.

$1, 1, x + 5, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x + 5$

Step 3

Multiply each term in $x + 9 + \frac{74}{x + 5} = y$ by $x + 5$ to eliminate the fractions.

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Step 3.1

Multiply each term in $x + 9 + \frac{74}{x + 5} = y$ by $x + 5$ .

$x (x + 5) + 9 (x + 5) + \frac{74}{x + 5} (x + 5) = y (x + 5)$

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

Apply the distributive property.

$x \cdot x + x \cdot 5 + 9 (x + 5) + \frac{74}{x + 5} (x + 5) = y (x + 5)$

Step 3.2.1.2

Multiply $x$ by $x$ .

$x^{2} + x \cdot 5 + 9 (x + 5) + \frac{74}{x + 5} (x + 5) = y (x + 5)$

Step 3.2.1.3

Move $5$ to the left of $x$ .

$x^{2} + 5 \cdot x + 9 (x + 5) + \frac{74}{x + 5} (x + 5) = y (x + 5)$

Step 3.2.1.4

Apply the distributive property.

$x^{2} + 5 x + 9 x + 9 \cdot 5 + \frac{74}{x + 5} (x + 5) = y (x + 5)$

Step 3.2.1.5

Multiply $9$ by $5$ .

$x^{2} + 5 x + 9 x + 45 + \frac{74}{x + 5} (x + 5) = y (x + 5)$

Step 3.2.1.6

Cancel the common factor of $x + 5$ .

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Step 3.2.1.6.1

Cancel the common factor.

$x^{2} + 5 x + 9 x + 45 + \frac{74}{x + 5} (x + 5) = y (x + 5)$

Step 3.2.1.6.2

Rewrite the expression.

$x^{2} + 5 x + 9 x + 45 + 74 = y (x + 5)$

Step 3.2.2

Simplify by adding terms.

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Step 3.2.2.1

Add $5 x$ and $9 x$ .

$x^{2} + 14 x + 45 + 74 = y (x + 5)$

Step 3.2.2.2

Add $45$ and $74$ .

$x^{2} + 14 x + 119 = y (x + 5)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$x^{2} + 14 x + 119 = y x + y \cdot 5$

Step 3.3.2

Move $5$ to the left of $y$ .

$x^{2} + 14 x + 119 = y x + 5 y$

Step 4

Solve the equation.

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Step 4.1

Subtract $y x$ from both sides of the equation.

$x^{2} + 14 x + 119 - y x = 5 y$

Step 4.2

Subtract $5 y$ from both sides of the equation.

$x^{2} + 14 x + 119 - y x - 5 y = 0$

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 $a = 1$ , $b = 14 - y$ , and $c = 119 - 5 y$ into the quadratic formula and solve for $x$ .

$\frac{- (14 - y) \pm \sqrt{{(14 - y)}^{2} - 4 \cdot (1 \cdot (119 - 5 y))}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Apply the distributive property.

$x = \frac{- 1 \cdot 14 + y \pm \sqrt{{(14 - y)}^{2} - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 1$ by $14$ .

$x = \frac{- 14 + y \pm \sqrt{{(14 - y)}^{2} - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.3

Multiply $- - y$ .

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Step 4.5.1.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{- 14 + 1 y \pm \sqrt{{(14 - y)}^{2} - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.3.2

Multiply $y$ by $1$ .

$x = \frac{- 14 + y \pm \sqrt{{(14 - y)}^{2} - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite ${(14 - y)}^{2}$ as $(14 - y) (14 - y)$ .

$x = \frac{- 14 + y \pm \sqrt{(14 - y) (14 - y) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.5

Expand $(14 - y) (14 - y)$ using the FOIL Method.

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Step 4.5.1.5.1

Apply the distributive property.

$x = \frac{- 14 + y \pm \sqrt{14 (14 - y) - y (14 - y) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.5.2

Apply the distributive property.

$x = \frac{- 14 + y \pm \sqrt{14 \cdot 14 + 14 (- y) - y (14 - y) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.5.3

Apply the distributive property.

$x = \frac{- 14 + y \pm \sqrt{14 \cdot 14 + 14 (- y) - y \cdot 14 - y (- y) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6

Simplify and combine like terms.

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Step 4.5.1.6.1

Simplify each term.

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Step 4.5.1.6.1.1

Multiply $14$ by $14$ .

$x = \frac{- 14 + y \pm \sqrt{196 + 14 (- y) - y \cdot 14 - y (- y) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6.1.2

Multiply $- 1$ by $14$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 14 y - y \cdot 14 - y (- y) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6.1.3

Multiply $14$ by $- 1$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 14 y - 14 y - y (- y) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6.1.4

Rewrite using the commutative property of multiplication.

$x = \frac{- 14 + y \pm \sqrt{196 - 14 y - 14 y - 1 \cdot (- 1 y \cdot y) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6.1.5

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

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Step 4.5.1.6.1.5.1

Move $y$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 14 y - 14 y - 1 \cdot (- 1 (y \cdot y)) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6.1.5.2

Multiply $y$ by $y$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 14 y - 14 y - 1 \cdot (- 1 y^{2}) - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6.1.6

Multiply $- 1$ by $- 1$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 14 y - 14 y + 1 y^{2} - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6.1.7

Multiply $y^{2}$ by $1$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 14 y - 14 y + y^{2} - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.6.2

Subtract $14 y$ from $- 14 y$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 28 y + y^{2} - 4 \cdot 1 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.7

Multiply $- 4$ by $1$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 28 y + y^{2} - 4 \cdot (119 - 5 y)}}{2 \cdot 1}$

Step 4.5.1.8

Apply the distributive property.

$x = \frac{- 14 + y \pm \sqrt{196 - 28 y + y^{2} - 4 \cdot 119 - 4 (- 5 y)}}{2 \cdot 1}$

Step 4.5.1.9

Multiply $- 4$ by $119$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 28 y + y^{2} - 476 - 4 (- 5 y)}}{2 \cdot 1}$

Step 4.5.1.10

Multiply $- 5$ by $- 4$ .

$x = \frac{- 14 + y \pm \sqrt{196 - 28 y + y^{2} - 476 + 20 y}}{2 \cdot 1}$

Step 4.5.1.11

Subtract $476$ from $196$ .

$x = \frac{- 14 + y \pm \sqrt{- 28 y + y^{2} - 280 + 20 y}}{2 \cdot 1}$

Step 4.5.1.12

Add $- 28 y$ and $20 y$ .

$x = \frac{- 14 + y \pm \sqrt{- 8 y + y^{2} - 280}}{2 \cdot 1}$

Step 4.5.1.13

Reorder terms.

$x = \frac{- 14 + y \pm \sqrt{y^{2} - 8 y - 280}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{- 14 + y \pm \sqrt{y^{2} - 8 y - 280}}{2}$

Step 4.6

The final answer is the combination of both solutions.

$x = - \frac{14 - y - \sqrt{y^{2} - 8 y - 280}}{2}$

$x = - \frac{14 - y + \sqrt{y^{2} - 8 y - 280}}{2}$

$x = - \frac{14 - y - \sqrt{y^{2} - 8 y - 280}}{2}$

$x = - \frac{14 - y + \sqrt{y^{2} - 8 y - 280}}{2}$