Trigonometry Examples

$\frac{x + 7}{x - 3} \cdot (f o r x) = 8$

Step 1

Factor each term.

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Step 1.1

Multiply $\frac{x + 7}{x - 3}$ by $f o r x$ .

$\frac{x + 7}{x - 3} (f o r x) = 8$

Step 1.2

Combine $f$ and $\frac{x + 7}{x - 3}$ .

$\frac{f (x + 7)}{x - 3} (o r x) = 8$

Step 1.3

Combine $o$ and $\frac{f (x + 7)}{x - 3}$ .

$\frac{o (f (x + 7))}{x - 3} (r x) = 8$

Step 1.4

Combine $r$ and $\frac{o (f (x + 7))}{x - 3}$ .

$\frac{r (o (f (x + 7)))}{x - 3} x = 8$

Step 1.5

Combine $\frac{r (o (f (x + 7)))}{x - 3}$ and $x$ .

$\frac{r (o (f (x + 7))) x}{x - 3} = 8$

Step 1.6

Remove parentheses.

$\frac{r (o f (x + 7)) x}{x - 3} = 8$

Step 1.7

Remove parentheses.

$\frac{r (o f) (x + 7) x}{x - 3} = 8$

Step 1.8

Remove parentheses.

$\frac{r o f (x + 7) x}{x - 3} = 8$

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.

$x - 3, 1$

Step 2.2

Remove parentheses.

$x - 3, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x - 3$

Step 3

Multiply each term in $\frac{r o f (x + 7) x}{x - 3} = 8$ by $x - 3$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{r o f (x + 7) x}{x - 3} = 8$ by $x - 3$ .

$\frac{r o f (x + 7) x}{x - 3} (x - 3) = 8 (x - 3)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify terms.

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Step 3.2.1.1

Cancel the common factor of $x - 3$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{r o f (x + 7) x}{x - 3} (x - 3) = 8 (x - 3)$

Step 3.2.1.1.2

Rewrite the expression.

$r o f (x + 7) x = 8 (x - 3)$

Step 3.2.1.2

Apply the distributive property.

$(r o f x + r o f \cdot 7) x = 8 (x - 3)$

Step 3.2.1.3

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

$(r o f x + 7 \cdot (r o f)) x = 8 (x - 3)$

Step 3.2.2

Remove parentheses.

$(r o f x + 7 r o f) x = 8 (x - 3)$

Step 3.2.3

Apply the distributive property.

$r o f x \cdot x + 7 r o f x = 8 (x - 3)$

Step 3.2.4

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

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Step 3.2.4.1

Move $x$ .

$r o f (x \cdot x) + 7 r o f x = 8 (x - 3)$

Step 3.2.4.2

Multiply $x$ by $x$ .

$r o f x^{2} + 7 r o f x = 8 (x - 3)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$r o f x^{2} + 7 r o f x = 8 x + 8 \cdot - 3$

Step 3.3.2

Multiply $8$ by $- 3$ .

$r o f x^{2} + 7 r o f x = 8 x - 24$

Step 4

Solve the equation.

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Step 4.1

Subtract $8 x$ from both sides of the equation.

$r o f x^{2} + 7 r o f x - 8 x = - 24$

Step 4.2

Add $24$ to both sides of the equation.

$r o f x^{2} + 7 r o f x - 8 x + 24 = 0$

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 $a = r o f$ , $b = 7 r o f - 8$ , and $c = 24$ into the quadratic formula and solve for $x$ .

$\frac{- (7 r o f - 8) \pm \sqrt{{(7 r o f - 8)}^{2} - 4 \cdot (r o f \cdot 24)}}{2 (r o f)}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Apply the distributive property.

$x = \frac{- (7 r o f) + 8 \pm \sqrt{{(7 r o f - 8)}^{2} - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.2

Multiply $7$ by $- 1$ .

$x = \frac{- 7 (r o f) + 8 \pm \sqrt{{(7 r o f - 8)}^{2} - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.3

Multiply $- 1$ by $- 8$ .

$x = \frac{- 7 (r o f) + 8 \pm \sqrt{{(7 r o f - 8)}^{2} - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.4

Remove parentheses.

$x = \frac{- 7 r o f + 8 \pm \sqrt{{(7 r o f - 8)}^{2} - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.5

Rewrite ${(7 r o f - 8)}^{2}$ as $(7 r o f - 8) (7 r o f - 8)$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{(7 r o f - 8) (7 r o f - 8) - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.6

Expand $(7 r o f - 8) (7 r o f - 8)$ using the FOIL Method.

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Step 4.5.6.1

Apply the distributive property.

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 r o f (7 r o f - 8) - 8 (7 r o f - 8) - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.6.2

Apply the distributive property.

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 r o f (7 r o f) + 7 r o f \cdot - 8 - 8 (7 r o f - 8) - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.6.3

Apply the distributive property.

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 r o f (7 r o f) + 7 r o f \cdot - 8 - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7

Simplify and combine like terms.

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Step 4.5.7.1

Simplify each term.

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Step 4.5.7.1.1

Multiply $r$ by $r$ by adding the exponents.

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Step 4.5.7.1.1.1

Move $r$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 (r \cdot r) o f (7 o f) + 7 r o f \cdot - 8 - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.1.2

Multiply $r$ by $r$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 r^{2} o f (7 o f) + 7 r o f \cdot - 8 - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.2

Multiply $o$ by $o$ by adding the exponents.

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Step 4.5.7.1.2.1

Move $o$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 r^{2} (o \cdot o) f (7 f) + 7 r o f \cdot - 8 - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.2.2

Multiply $o$ by $o$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 r^{2} o^{2} f (7 f) + 7 r o f \cdot - 8 - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.3

Multiply $f$ by $f$ by adding the exponents.

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Step 4.5.7.1.3.1

Move $f$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 r^{2} o^{2} (f \cdot f) \cdot 7 + 7 r o f \cdot - 8 - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.3.2

Multiply $f$ by $f$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{7 r^{2} o^{2} f^{2} \cdot 7 + 7 r o f \cdot - 8 - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.4

Multiply $7$ by $7$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{49 r^{2} o^{2} f^{2} + 7 r o f \cdot - 8 - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.5

Multiply $- 8$ by $7$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{49 r^{2} o^{2} f^{2} - 56 r o f - 8 (7 r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.6

Multiply $7$ by $- 8$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{49 r^{2} o^{2} f^{2} - 56 r o f - 56 (r o f) - 8 \cdot - 8 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.1.7

Multiply $- 8$ by $- 8$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{49 r^{2} o^{2} f^{2} - 56 r o f - 56 r o f + 64 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.7.2

Subtract $56 r o f$ from $- 56 r o f$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{49 r^{2} o^{2} f^{2} - 112 r o f + 64 - 4 \cdot (r o f) \cdot 24}}{2 r o f}$

Step 4.5.8

Multiply $24$ by $- 4$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{49 r^{2} o^{2} f^{2} - 112 r o f + 64 - 96 r o f}}{2 r o f}$

Step 4.5.9

Subtract $96 r o f$ from $- 112 r o f$ .

$x = \frac{- 7 r o f + 8 \pm \sqrt{49 r^{2} o^{2} f^{2} - 208 r o f + 64}}{2 r o f}$

Step 4.6

The final answer is the combination of both solutions.

$x = - \frac{7 r o f - 8 - \sqrt{49 r^{2} o^{2} f^{2} - 208 r o f + 64}}{2 r o f}$

$x = - \frac{7 r o f - 8 + \sqrt{49 r^{2} o^{2} f^{2} - 208 r o f + 64}}{2 r o f}$

$x = - \frac{7 r o f - 8 - \sqrt{49 r^{2} o^{2} f^{2} - 208 r o f + 64}}{2 r o f}$

$x = - \frac{7 r o f - 8 + \sqrt{49 r^{2} o^{2} f^{2} - 208 r o f + 64}}{2 r o f}$