Trigonometry Examples

$\frac{5}{x + 4} - 4 = \frac{9 x + 16}{x + 6}$

Step 1

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

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Step 1.1

Add $4$ to both sides of the equation.

$\frac{5}{x + 4} = \frac{9 x + 16}{x + 6} + 4$

Step 1.2

Split the fraction $\frac{9 x + 16}{x + 6}$ into two fractions.

$\frac{5}{x + 4} = \frac{9 x}{x + 6} + \frac{16}{x + 6} + 4$

Step 1.3

Find the common denominator.

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Step 1.3.1

Write $4$ as a fraction with denominator $1$ .

$\frac{5}{x + 4} = \frac{9 x}{x + 6} + \frac{16}{x + 6} + \frac{4}{1}$

Step 1.3.2

Multiply $\frac{4}{1}$ by $\frac{x + 6}{x + 6}$ .

$\frac{5}{x + 4} = \frac{9 x}{x + 6} + \frac{16}{x + 6} + \frac{4}{1} \cdot \frac{x + 6}{x + 6}$

Step 1.3.3

Multiply $\frac{4}{1}$ by $\frac{x + 6}{x + 6}$ .

$\frac{5}{x + 4} = \frac{9 x}{x + 6} + \frac{16}{x + 6} + \frac{4 (x + 6)}{x + 6}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{5}{x + 4} = \frac{9 x + 16 + 4 (x + 6)}{x + 6}$

Step 1.5

Simplify each term.

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Step 1.5.1

Apply the distributive property.

$\frac{5}{x + 4} = \frac{9 x + 16 + 4 x + 4 \cdot 6}{x + 6}$

Step 1.5.2

Multiply $4$ by $6$ .

$\frac{5}{x + 4} = \frac{9 x + 16 + 4 x + 24}{x + 6}$

Step 1.6

Add $9 x$ and $4 x$ .

$\frac{5}{x + 4} = \frac{13 x + 16 + 24}{x + 6}$

Step 1.7

Add $16$ and $24$ .

$\frac{5}{x + 4} = \frac{13 x + 40}{x + 6}$

Step 1.8

Split the fraction $\frac{13 x + 40}{x + 6}$ into two fractions.

$\frac{5}{x + 4} = \frac{13 x}{x + 6} + \frac{40}{x + 6}$

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.

$x + 4, x + 6, x + 6$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.5

The factor for $x + 4$ is $x + 4$ itself.

$(x + 4) = x + 4$

$(x + 4)$ occurs $1$ time.

Step 2.6

The factor for $x + 6$ is $x + 6$ itself.

$(x + 6) = x + 6$

$(x + 6)$ occurs $1$ time.

Step 2.7

The LCM of $x + 4, x + 6, x + 6$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x + 4) (x + 6)$

Step 3

Multiply each term in $\frac{5}{x + 4} = \frac{13 x}{x + 6} + \frac{40}{x + 6}$ by $(x + 4) (x + 6)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{5}{x + 4} = \frac{13 x}{x + 6} + \frac{40}{x + 6}$ by $(x + 4) (x + 6)$ .

$\frac{5}{x + 4} ((x + 4) (x + 6)) = \frac{13 x}{x + 6} ((x + 4) (x + 6)) + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x + 4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5}{x + 4} ((x + 4) (x + 6)) = \frac{13 x}{x + 6} ((x + 4) (x + 6)) + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.2.1.2

Rewrite the expression.

$5 (x + 6) = \frac{13 x}{x + 6} ((x + 4) (x + 6)) + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.2.2

Apply the distributive property.

$5 x + 5 \cdot 6 = \frac{13 x}{x + 6} ((x + 4) (x + 6)) + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.2.3

Multiply $5$ by $6$ .

$5 x + 30 = \frac{13 x}{x + 6} ((x + 4) (x + 6)) + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $x + 6$ .

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Step 3.3.1.1.1

Factor $x + 6$ out of $(x + 4) (x + 6)$ .

$5 x + 30 = \frac{13 x}{x + 6} ((x + 6) (x + 4)) + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.3.1.1.2

Cancel the common factor.

$5 x + 30 = \frac{13 x}{x + 6} ((x + 6) (x + 4)) + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.3.1.1.3

Rewrite the expression.

$5 x + 30 = 13 x (x + 4) + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.3.1.2

Apply the distributive property.

$5 x + 30 = 13 x \cdot x + 13 x \cdot 4 + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.3.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.3.1

Move $x$ .

$5 x + 30 = 13 (x \cdot x) + 13 x \cdot 4 + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.3.1.3.2

Multiply $x$ by $x$ .

$5 x + 30 = 13 x^{2} + 13 x \cdot 4 + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.3.1.4

Multiply $4$ by $13$ .

$5 x + 30 = 13 x^{2} + 52 x + \frac{40}{x + 6} ((x + 4) (x + 6))$

Step 3.3.1.5

Cancel the common factor of $x + 6$ .

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Step 3.3.1.5.1

Factor $x + 6$ out of $(x + 4) (x + 6)$ .

$5 x + 30 = 13 x^{2} + 52 x + \frac{40}{x + 6} ((x + 6) (x + 4))$

Step 3.3.1.5.2

Cancel the common factor.

$5 x + 30 = 13 x^{2} + 52 x + \frac{40}{x + 6} ((x + 6) (x + 4))$

Step 3.3.1.5.3

Rewrite the expression.

$5 x + 30 = 13 x^{2} + 52 x + 40 (x + 4)$

Step 3.3.1.6

Apply the distributive property.

$5 x + 30 = 13 x^{2} + 52 x + 40 x + 40 \cdot 4$

Step 3.3.1.7

Multiply $40$ by $4$ .

$5 x + 30 = 13 x^{2} + 52 x + 40 x + 160$

Step 3.3.2

Add $52 x$ and $40 x$ .

$5 x + 30 = 13 x^{2} + 92 x + 160$

Step 4

Solve the equation.

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Step 4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$13 x^{2} + 92 x + 160 = 5 x + 30$

Step 4.2

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

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Step 4.2.1

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

$13 x^{2} + 92 x + 160 - 5 x = 30$

Step 4.2.2

Subtract $5 x$ from $92 x$ .

$13 x^{2} + 87 x + 160 = 30$

Step 4.3

Move all terms to the left side of the equation and simplify.

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Step 4.3.1

Subtract $30$ from both sides of the equation.

$13 x^{2} + 87 x + 160 - 30 = 0$

Step 4.3.2

Subtract $30$ from $160$ .

$13 x^{2} + 87 x + 130 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = 13$ , $b = 87$ , and $c = 130$ into the quadratic formula and solve for $x$ .

$\frac{- 87 \pm \sqrt{87^{2} - 4 \cdot (13 \cdot 130)}}{2 \cdot 13}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $87$ to the power of $2$ .

$x = \frac{- 87 \pm \sqrt{7569 - 4 \cdot 13 \cdot 130}}{2 \cdot 13}$

Step 4.6.1.2

Multiply $- 4 \cdot 13 \cdot 130$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $13$ .

$x = \frac{- 87 \pm \sqrt{7569 - 52 \cdot 130}}{2 \cdot 13}$

Step 4.6.1.2.2

Multiply $- 52$ by $130$ .

$x = \frac{- 87 \pm \sqrt{7569 - 6760}}{2 \cdot 13}$

Step 4.6.1.3

Subtract $6760$ from $7569$ .

$x = \frac{- 87 \pm \sqrt{809}}{2 \cdot 13}$

Step 4.6.2

Multiply $2$ by $13$ .

$x = \frac{- 87 \pm \sqrt{809}}{26}$

Step 4.7

The final answer is the combination of both solutions.

$x = - \frac{87 - \sqrt{809}}{26}, - \frac{87 + \sqrt{809}}{26}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{87 - \sqrt{809}}{26}, - \frac{87 + \sqrt{809}}{26}$

Decimal Form:

$x = - 2.25219518 \dots, - 4.44011251 \dots$