Trigonometry Examples

$\frac{x^{2} - 7}{x + 4} = f (x)$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$x + 4, 1$

Step 1.2

Remove parentheses.

$x + 4, 1$

Step 1.3

The LCM of one and any expression is the expression.

$x + 4$

Step 2

Multiply each term in $\frac{x^{2} - 7}{x + 4} = f (x)$ by $x + 4$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x^{2} - 7}{x + 4} = f (x)$ by $x + 4$ .

$\frac{x^{2} - 7}{x + 4} (x + 4) = f (x) (x + 4)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x + 4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{x^{2} - 7}{x + 4} (x + 4) = f (x) (x + 4)$

Step 2.2.1.2

Rewrite the expression.

$x^{2} - 7 = f (x) (x + 4)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Apply the distributive property.

$x^{2} - 7 = f (x) x + f (x) \cdot 4$

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

Move $4$ to the left of $f (x)$ .

$x^{2} - 7 = f (x) x + 4 \cdot f (x)$

Step 2.3.2.2

Reorder factors in $f (x) x + 4 f (x)$ .

$x^{2} - 7 = x f (x) + 4 f (x)$

Step 3

Solve the equation.

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Step 3.1

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

$x f (x) + 4 f x = x^{2} - 7$

Step 3.2

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

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Step 3.2.1

Move $x$ .

$x \cdot x f + 4 f x = x^{2} - 7$

Step 3.2.2

Multiply $x$ by $x$ .

$x^{2} f + 4 f x = x^{2} - 7$

Step 3.3

Subtract $x^{2}$ from both sides of the equation.

$x^{2} f + 4 f x - x^{2} = - 7$

Step 3.4

Add $7$ to both sides of the equation.

$x^{2} f + 4 f x - x^{2} + 7 = 0$

Step 3.5

Use the quadratic formula to find the solutions.

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

Step 3.6

Substitute the values $a = f - 1$ , $b = 4 f$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{- 4 f \pm \sqrt{{(4 f)}^{2} - 4 \cdot ((f - 1) \cdot 7)}}{2 (f - 1)}$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Add parentheses.

$x = \frac{- 4 f \pm \sqrt{{(4 f)}^{2} - 4 \cdot ((f - 1) \cdot 7)}}{2 (f - 1)}$

Step 3.7.1.2

Let $u = (f - 1) \cdot 7$ . Substitute $u$ for all occurrences of $(f - 1) \cdot 7$ .

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Step 3.7.1.2.1

Apply the product rule to $4 f$ .

$x = \frac{- 4 f \pm \sqrt{4^{2} f^{2} - 4 \cdot u}}{2 (f - 1)}$

Step 3.7.1.2.2

Raise $4$ to the power of $2$ .

$x = \frac{- 4 f \pm \sqrt{16 f^{2} - 4 u}}{2 (f - 1)}$

Step 3.7.1.3

Factor $4$ out of $16 f^{2} - 4 u$ .

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Step 3.7.1.3.1

Factor $4$ out of $16 f^{2}$ .

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2}) - 4 u}}{2 (f - 1)}$

Step 3.7.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2}) + 4 (- u)}}{2 (f - 1)}$

Step 3.7.1.3.3

Factor $4$ out of $4 (4 f^{2}) + 4 (- u)$ .

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2} - u)}}{2 (f - 1)}$

Step 3.7.1.4

Replace all occurrences of $u$ with $(f - 1) \cdot 7$ .

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2} - ((f - 1) \cdot 7))}}{2 (f - 1)}$

Step 3.7.1.5

Simplify each term.

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Step 3.7.1.5.1

Apply the distributive property.

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2} - (f \cdot 7 - 1 \cdot 7))}}{2 (f - 1)}$

Step 3.7.1.5.2

Move $7$ to the left of $f$ .

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2} - (7 \cdot f - 1 \cdot 7))}}{2 (f - 1)}$

Step 3.7.1.5.3

Multiply $- 1$ by $7$ .

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2} - (7 \cdot f - 7))}}{2 (f - 1)}$

Step 3.7.1.5.4

Apply the distributive property.

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2} - (7 f) + 7)}}{2 (f - 1)}$

Step 3.7.1.5.5

Multiply $7$ by $- 1$ .

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2} - 7 f + 7)}}{2 (f - 1)}$

Step 3.7.1.5.6

Multiply $- 1$ by $- 7$ .

$x = \frac{- 4 f \pm \sqrt{4 (4 f^{2} - 7 f + 7)}}{2 (f - 1)}$

Step 3.7.1.6

Rewrite $4 (4 f^{2} - 7 f + 7)$ as $2^{2} ({(2 f)}^{2} - 7 f + 7)$ .

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Step 3.7.1.6.1

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

$x = \frac{- 4 f \pm \sqrt{2^{2} (4 f^{2} - 7 f + 7)}}{2 (f - 1)}$

Step 3.7.1.6.2

Rewrite $4 f^{2}$ as ${(2 f)}^{2}$ .

$x = \frac{- 4 f \pm \sqrt{2^{2} ({(2 f)}^{2} - 7 f + 7)}}{2 (f - 1)}$

Step 3.7.1.7

Pull terms out from under the radical.

$x = \frac{- 4 f \pm 2 \sqrt{{(2 f)}^{2} - 7 f + 7}}{2 (f - 1)}$

Step 3.7.1.8

Apply the product rule to $2 f$ .

$x = \frac{- 4 f \pm 2 \sqrt{2^{2} f^{2} - 7 f + 7}}{2 (f - 1)}$

Step 3.7.1.9

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

$x = \frac{- 4 f \pm 2 \sqrt{4 f^{2} - 7 f + 7}}{2 (f - 1)}$

Step 3.7.2

Simplify $\frac{- 4 f \pm 2 \sqrt{4 f^{2} - 7 f + 7}}{2 (f - 1)}$ .

$x = \frac{- 2 f \pm \sqrt{4 f^{2} - 7 f + 7}}{f - 1}$

Step 3.8

The final answer is the combination of both solutions.

$x = - \frac{2 f - \sqrt{4 f^{2} - 7 f + 7}}{f - 1}$

$x = - \frac{2 f + \sqrt{4 f^{2} - 7 f + 7}}{f - 1}$

$x = - \frac{2 f - \sqrt{4 f^{2} - 7 f + 7}}{f - 1}$

$x = - \frac{2 f + \sqrt{4 f^{2} - 7 f + 7}}{f - 1}$