Trigonometry Examples

$\frac{a^{2} - 4 a - 12}{a^{2} - 10 a + 25} = \frac{6}{a - 5} + \frac{a - 3}{a - 5}$

Step 1

Factor each term.

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Step 1.1

Factor $a^{2} - 4 a - 12$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{(a - 6) (a + 2)}{a^{2} - 10 a + 25} = \frac{6}{a - 5} + \frac{a - 3}{a - 5}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{(a - 6) (a + 2)}{a^{2} - 10 a + 5^{2}} = \frac{6}{a - 5} + \frac{a - 3}{a - 5}$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 a = 2 \cdot a \cdot 5$

Step 1.2.3

Rewrite the polynomial.

$\frac{(a - 6) (a + 2)}{a^{2} - 2 \cdot a \cdot 5 + 5^{2}} = \frac{6}{a - 5} + \frac{a - 3}{a - 5}$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 5$ .

$\frac{(a - 6) (a + 2)}{{(a - 5)}^{2}} = \frac{6}{a - 5} + \frac{a - 3}{a - 5}$

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.

${(a - 5)}^{2}, a - 5, a - 5$

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 factors for $a - 5$ are $(a - 5) \cdot (a - 5)$ , which is $a - 5$ multiplied by itself $2$ times.

$(a - 5) = (a - 5) \cdot (a - 5)$

$(a - 5)$ occurs $2$ times.

Step 2.6

The factor for $a - 5$ is $a - 5$ itself.

$(a - 5) = a - 5$

$(a - 5)$ occurs $1$ time.

Step 2.7

The LCM of ${(a - 5)}^{2}, a - 5, a - 5$ is the result of multiplying all factors the greatest number of times they occur in either term.

${(a - 5)}^{2}$

Step 3

Multiply each term in $\frac{(a - 6) (a + 2)}{{(a - 5)}^{2}} = \frac{6}{a - 5} + \frac{a - 3}{a - 5}$ by ${(a - 5)}^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{(a - 6) (a + 2)}{{(a - 5)}^{2}} = \frac{6}{a - 5} + \frac{a - 3}{a - 5}$ by ${(a - 5)}^{2}$ .

$\frac{(a - 6) (a + 2)}{{(a - 5)}^{2}} {(a - 5)}^{2} = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of ${(a - 5)}^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{(a - 6) (a + 2)}{{(a - 5)}^{2}} {(a - 5)}^{2} = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2.1.2

Rewrite the expression.

$(a - 6) (a + 2) = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2.2

Expand $(a - 6) (a + 2)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$a (a + 2) - 6 (a + 2) = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2.2.2

Apply the distributive property.

$a \cdot a + a \cdot 2 - 6 (a + 2) = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2.2.3

Apply the distributive property.

$a \cdot a + a \cdot 2 - 6 a - 6 \cdot 2 = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply $a$ by $a$ .

$a^{2} + a \cdot 2 - 6 a - 6 \cdot 2 = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2.3.1.2

Move $2$ to the left of $a$ .

$a^{2} + 2 \cdot a - 6 a - 6 \cdot 2 = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2.3.1.3

Multiply $- 6$ by $2$ .

$a^{2} + 2 a - 6 a - 12 = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.2.3.2

Subtract $6 a$ from $2 a$ .

$a^{2} - 4 a - 12 = \frac{6}{a - 5} {(a - 5)}^{2} + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

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 $a - 5$ .

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Step 3.3.1.1.1

Factor $a - 5$ out of ${(a - 5)}^{2}$ .

$a^{2} - 4 a - 12 = \frac{6}{a - 5} ((a - 5) (a - 5)) + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.3.1.1.2

Cancel the common factor.

$a^{2} - 4 a - 12 = \frac{6}{a - 5} ((a - 5) (a - 5)) + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.3.1.1.3

Rewrite the expression.

$a^{2} - 4 a - 12 = 6 (a - 5) + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.3.1.2

Apply the distributive property.

$a^{2} - 4 a - 12 = 6 a + 6 \cdot - 5 + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.3.1.3

Multiply $6$ by $- 5$ .

$a^{2} - 4 a - 12 = 6 a - 30 + \frac{a - 3}{a - 5} {(a - 5)}^{2}$

Step 3.3.1.4

Cancel the common factor of $a - 5$ .

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Step 3.3.1.4.1

Factor $a - 5$ out of ${(a - 5)}^{2}$ .

$a^{2} - 4 a - 12 = 6 a - 30 + \frac{a - 3}{a - 5} ((a - 5) (a - 5))$

Step 3.3.1.4.2

Cancel the common factor.

$a^{2} - 4 a - 12 = 6 a - 30 + \frac{a - 3}{a - 5} ((a - 5) (a - 5))$

Step 3.3.1.4.3

Rewrite the expression.

$a^{2} - 4 a - 12 = 6 a - 30 + (a - 3) (a - 5)$

Step 3.3.1.5

Expand $(a - 3) (a - 5)$ using the FOIL Method.

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Step 3.3.1.5.1

Apply the distributive property.

$a^{2} - 4 a - 12 = 6 a - 30 + a (a - 5) - 3 (a - 5)$

Step 3.3.1.5.2

Apply the distributive property.

$a^{2} - 4 a - 12 = 6 a - 30 + a \cdot a + a \cdot - 5 - 3 (a - 5)$

Step 3.3.1.5.3

Apply the distributive property.

$a^{2} - 4 a - 12 = 6 a - 30 + a \cdot a + a \cdot - 5 - 3 a - 3 \cdot - 5$

Step 3.3.1.6

Simplify and combine like terms.

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Step 3.3.1.6.1

Simplify each term.

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Step 3.3.1.6.1.1

Multiply $a$ by $a$ .

$a^{2} - 4 a - 12 = 6 a - 30 + a^{2} + a \cdot - 5 - 3 a - 3 \cdot - 5$

Step 3.3.1.6.1.2

Move $- 5$ to the left of $a$ .

$a^{2} - 4 a - 12 = 6 a - 30 + a^{2} - 5 \cdot a - 3 a - 3 \cdot - 5$

Step 3.3.1.6.1.3

Multiply $- 3$ by $- 5$ .

$a^{2} - 4 a - 12 = 6 a - 30 + a^{2} - 5 a - 3 a + 15$

Step 3.3.1.6.2

Subtract $3 a$ from $- 5 a$ .

$a^{2} - 4 a - 12 = 6 a - 30 + a^{2} - 8 a + 15$

Step 3.3.2

Simplify by adding terms.

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Step 3.3.2.1

Subtract $8 a$ from $6 a$ .

$a^{2} - 4 a - 12 = - 2 a - 30 + a^{2} + 15$

Step 3.3.2.2

Add $- 30$ and $15$ .

$a^{2} - 4 a - 12 = - 2 a + a^{2} - 15$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

Add $2 a$ to both sides of the equation.

$a^{2} - 4 a - 12 + 2 a = a^{2} - 15$

Step 4.1.2

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

$a^{2} - 4 a - 12 + 2 a - a^{2} = - 15$

Step 4.1.3

Combine the opposite terms in $a^{2} - 4 a - 12 + 2 a - a^{2}$ .

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Step 4.1.3.1

Subtract $a^{2}$ from $a^{2}$ .

$- 4 a - 12 + 2 a + 0 = - 15$

Step 4.1.3.2

Add $- 4 a - 12 + 2 a$ and $0$ .

$- 4 a - 12 + 2 a = - 15$

Step 4.1.4

Add $- 4 a$ and $2 a$ .

$- 2 a - 12 = - 15$

Step 4.2

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

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Step 4.2.1

Add $12$ to both sides of the equation.

$- 2 a = - 15 + 12$

Step 4.2.2

Add $- 15$ and $12$ .

$- 2 a = - 3$

Step 4.3

Divide each term in $- 2 a = - 3$ by $- 2$ and simplify.

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Step 4.3.1

Divide each term in $- 2 a = - 3$ by $- 2$ .

$\frac{- 2 a}{- 2} = \frac{- 3}{- 2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 2 a}{- 2} = \frac{- 3}{- 2}$

Step 4.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 3}{- 2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Dividing two negative values results in a positive value.

$a = \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$a = \frac{3}{2}$

Decimal Form:

$a = 1.5$

Mixed Number Form:

$a = 1 \frac{1}{2}$