Basic Math Examples

$\frac{x + 5}{6 - x} - \frac{x^{2} - 11 x + 28}{x^{2} - 13 x + 42}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 11 x + 28$ 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 $28$ and whose sum is $- 11$ .

$- 7, - 4$

Step 1.1.2

Write the factored form using these integers.

$\frac{x + 5}{6 - x} - \frac{(x - 7) (x - 4)}{x^{2} - 13 x + 42}$

Step 1.2

Factor $x^{2} - 13 x + 42$ using the AC method.

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Step 1.2.1

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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Step 1.2.2

Write the factored form using these integers.

$\frac{x + 5}{6 - x} - \frac{(x - 7) (x - 4)}{(x - 7) (x - 6)}$

Step 1.3

Cancel the common factor of $x - 7$ .

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Step 1.3.1

Cancel the common factor.

$\frac{x + 5}{6 - x} - \frac{(x - 7) (x - 4)}{(x - 7) (x - 6)}$

Step 1.3.2

Rewrite the expression.

$\frac{x + 5}{6 - x} - \frac{x - 4}{x - 6}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $x$ .

$\frac{x + 5}{6 - x} - \frac{x - 4}{- 1 (- x) - 6}$

Step 2.2

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{x + 5}{6 - x} - \frac{x - 4}{- 1 (- x) - 1 (6)}$

Step 2.3

Factor $- 1$ out of $- 1 (- x) - 1 (6)$ .

$\frac{x + 5}{6 - x} - \frac{x - 4}{- 1 (- x + 6)}$

Step 2.4

Reorder terms.

$\frac{x + 5}{- x + 6} - \frac{x - 4}{- 1 (- x + 6)}$

Step 3

To write $\frac{x + 5}{- x + 6}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{x + 5}{- x + 6} \cdot \frac{- 1}{- 1} - \frac{x - 4}{- 1 (- x + 6)}$

Step 4

Write each expression with a common denominator of $(- x + 6) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x + 5}{- x + 6}$ by $\frac{- 1}{- 1}$ .

$\frac{(x + 5) \cdot - 1}{(- x + 6) \cdot - 1} - \frac{x - 4}{- 1 (- x + 6)}$

Step 4.2

Reorder the factors of $(- x + 6) \cdot - 1$ .

$\frac{(x + 5) \cdot - 1}{- (- x + 6)} - \frac{x - 4}{- 1 (- x + 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x + 5) \cdot - 1 - (x - 4)}{- (- x + 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot - 1 + 5 \cdot - 1 - (x - 4)}{- (- x + 6)}$

Step 6.2

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x + 5 \cdot - 1 - (x - 4)}{- (- x + 6)}$

Step 6.3

Multiply $5$ by $- 1$ .

$\frac{- 1 \cdot x - 5 - (x - 4)}{- (- x + 6)}$

Step 6.4

Rewrite $- 1 x$ as $- x$ .

$\frac{- x - 5 - (x - 4)}{- (- x + 6)}$

Step 6.5

Apply the distributive property.

$\frac{- x - 5 - x - - 4}{- (- x + 6)}$

Step 6.6

Multiply $- 1$ by $- 4$ .

$\frac{- x - 5 - x + 4}{- (- x + 6)}$

Step 6.7

Subtract $x$ from $- x$ .

$\frac{- 2 x - 5 + 4}{- (- x + 6)}$

Step 6.8

Add $- 5$ and $4$ .

$\frac{- 2 x - 1}{- (- x + 6)}$

Step 7

Simplify terms.

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Step 7.1

Move the negative in front of the fraction.

$- \frac{- 2 x - 1}{- x + 6}$

Step 7.2

Factor $- 1$ out of $- 2 x$ .

$- \frac{- (2 x) - 1}{- x + 6}$

Step 7.3

Rewrite $- 1$ as $- 1 (1)$ .

$- \frac{- (2 x) - 1 (1)}{- x + 6}$

Step 7.4

Factor $- 1$ out of $- (2 x) - 1 (1)$ .

$- \frac{- (2 x + 1)}{- x + 6}$

Step 7.5

Rewrite $- (2 x + 1)$ as $- 1 (2 x + 1)$ .

$- \frac{- 1 (2 x + 1)}{- x + 6}$

Step 7.6

Factor $- 1$ out of $- x$ .

$- \frac{- 1 (2 x + 1)}{- (x) + 6}$

Step 7.7

Rewrite $6$ as $- 1 (- 6)$ .

$- \frac{- 1 (2 x + 1)}{- (x) - 1 (- 6)}$

Step 7.8

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$- \frac{- 1 (2 x + 1)}{- (x - 6)}$

Step 7.9

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

$- \frac{- 1 (2 x + 1)}{- 1 (x - 6)}$

Step 7.10

Cancel the common factor.

$- \frac{- 1 (2 x + 1)}{- 1 (x - 6)}$

Step 7.11

Rewrite the expression.

$- \frac{2 x + 1}{x - 6}$