Basic Math Examples

\frac{r - 2}{r^{2} - 2 r - 8} + \frac{r - 3}{r + 2} = \frac{1}{r^{2} - 2 r - 8}

$\frac{r - 2}{r^{2} - 2 r - 8} + \frac{r - 3}{r + 2} = \frac{1}{r^{2} - 2 r - 8}$

Step 1

Factor each term.

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Step 1.1

Factor

r^{2} - 2 r - 8

$r^{2} - 2 r - 8$ using the AC method.

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Step 1.1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 8

$- 8$ and whose sum is

- 2

$- 2$ .

- 4, 2

$- 4, 2$

Step 1.1.2

Write the factored form using these integers.

\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{r^{2} - 2 r - 8}

$\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{r^{2} - 2 r - 8}$

\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{r^{2} - 2 r - 8}

$\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{r^{2} - 2 r - 8}$

Step 1.2

Factor

r^{2} - 2 r - 8

$r^{2} - 2 r - 8$ using the AC method.

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Step 1.2.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 8

$- 8$ and whose sum is

- 2

$- 2$ .

- 4, 2

$- 4, 2$

Step 1.2.2

Write the factored form using these integers.

\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{(r - 4) (r + 2)}

$\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{(r - 4) (r + 2)}$

\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{(r - 4) (r + 2)}

$\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{(r - 4) (r + 2)}$

\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{(r - 4) (r + 2)}

$\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{(r - 4) (r + 2)}$

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.

(r - 4) (r + 2), r + 2, (r - 4) (r + 2)

$(r - 4) (r + 2), r + 2, (r - 4) (r + 2)$

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

$1$ 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

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

1

$1$

Step 2.5

The factor for

r - 4

$r - 4$ is

r - 4

$r - 4$ itself.

(r - 4) = r - 4

$(r - 4) = r - 4$

(r - 4)

$(r - 4)$ occurs

1

$1$ time.

Step 2.6

The factor for

r + 2

$r + 2$ is

$r + 2$ itself.

$(r + 2) = r + 2$

$(r + 2)$ occurs

$1$ time.

Step 2.7

The factor for

$r - 4$ is

$r - 4$ itself.

$(r - 4) = r - 4$

$(r - 4)$ occurs

$1$ time.

Step 2.8

The factor for

$r + 2$ is

$r + 2$ itself.

$(r + 2) = r + 2$

$(r + 2)$ occurs

$1$ time.

Step 2.9

The LCM of

$r - 4, r + 2, r + 2, r - 4, r + 2$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(r - 4) (r + 2)$

Step 3

Multiply each term in

$\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{(r - 4) (r + 2)}$ by

$(r - 4) (r + 2)$ to eliminate the fractions.

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Step 3.1

Multiply each term in

$\frac{r - 2}{(r - 4) (r + 2)} + \frac{r - 3}{r + 2} = \frac{1}{(r - 4) (r + 2)}$ by

$(r - 4) (r + 2)$ .

$\frac{r - 2}{(r - 4) (r + 2)} ((r - 4) (r + 2)) + \frac{r - 3}{r + 2} ((r - 4) (r + 2)) = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

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

Cancel the common factor of

$(r - 4) (r + 2)$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{r - 2}{(r - 4) (r + 2)} ((r - 4) (r + 2)) + \frac{r - 3}{r + 2} ((r - 4) (r + 2)) = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.1.2

Rewrite the expression.

$r - 2 + \frac{r - 3}{r + 2} ((r - 4) (r + 2)) = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.2

Cancel the common factor of

$r + 2$ .

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Step 3.2.1.2.1

Factor

$r + 2$ out of

$(r - 4) (r + 2)$ .

$r - 2 + \frac{r - 3}{r + 2} ((r + 2) (r - 4)) = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.2.2

Cancel the common factor.

$r - 2 + \frac{r - 3}{r + 2} ((r + 2) (r - 4)) = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.2.3

Rewrite the expression.

$r - 2 + (r - 3) (r - 4) = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.3

Expand

$(r - 3) (r - 4)$ using the FOIL Method.

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Step 3.2.1.3.1

Apply the distributive property.

$r - 2 + r (r - 4) - 3 (r - 4) = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.3.2

Apply the distributive property.

$r - 2 + r \cdot r + r \cdot - 4 - 3 (r - 4) = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.3.3

Apply the distributive property.

$r - 2 + r \cdot r + r \cdot - 4 - 3 r - 3 \cdot - 4 = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.4

Simplify and combine like terms.

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Step 3.2.1.4.1

Simplify each term.

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Step 3.2.1.4.1.1

Multiply

$r$ by

$r$ .

$r - 2 + r^{2} + r \cdot - 4 - 3 r - 3 \cdot - 4 = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.4.1.2

Move

$- 4$ to the left of

$r$ .

$r - 2 + r^{2} - 4 \cdot r - 3 r - 3 \cdot - 4 = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.4.1.3

Multiply

$- 3$ by

$- 4$ .

$r - 2 + r^{2} - 4 r - 3 r + 12 = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.1.4.2

Subtract

$3 r$ from

$- 4 r$ .

$r - 2 + r^{2} - 7 r + 12 = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.2

Simplify by adding terms.

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Step 3.2.2.1

Subtract

$7 r$ from

$r$ .

$- 6 r - 2 + r^{2} + 12 = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.2.2.2

Add

$- 2$ and

$12$ .

$- 6 r + r^{2} + 10 = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of

$(r - 4) (r + 2)$ .

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Step 3.3.1.1

Cancel the common factor.

$- 6 r + r^{2} + 10 = \frac{1}{(r - 4) (r + 2)} ((r - 4) (r + 2))$

Step 3.3.1.2

Rewrite the expression.

$- 6 r + r^{2} + 10 = 1$

Step 4

Solve the equation.

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Step 4.1

Subtract

$1$ from both sides of the equation.

$- 6 r + r^{2} + 10 - 1 = 0$

Step 4.2

Subtract

$1$ from

$10$ .

$- 6 r + r^{2} + 9 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Let

$u = r$ . Substitute

$u$ for all occurrences of

$r$ .

$- 6 u + u^{2} + 9 = 0$

Step 4.3.2

Factor using the perfect square rule.

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Step 4.3.2.1

Rearrange terms.

$u^{2} - 6 u + 9 = 0$

Step 4.3.2.2

Rewrite

$9$ as

$3^{2}$ .

$u^{2} - 6 u + 3^{2} = 0$

Step 4.3.2.3

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

$6 u = 2 \cdot u \cdot 3$

Step 4.3.2.4

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 3 + 3^{2} = 0$

Step 4.3.2.5

Factor using the perfect square trinomial rule

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

$a = u$ and

$b = 3$ .

${(u - 3)}^{2} = 0$

Step 4.3.3

Replace all occurrences of

$u$ with

$r$ .

${(r - 3)}^{2} = 0$

Step 4.4

Set the

$r - 3$ equal to

$0$ .

$r - 3 = 0$

Step 4.5

Add

$3$ to both sides of the equation.

$r = 3$