Basic Math Examples

$\frac{j - 7}{j + 6} - \frac{2 j + 5}{j^{2} + 8 j + 12}$

Step 1

Factor $j^{2} + 8 j + 12$ using the AC method.

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Step 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 $8$ .

$2, 6$

Step 1.2

Write the factored form using these integers.

$\frac{j - 7}{j + 6} - \frac{2 j + 5}{(j + 2) (j + 6)}$

Step 2

To write $\frac{j - 7}{j + 6}$ as a fraction with a common denominator, multiply by $\frac{j + 2}{j + 2}$ .

$\frac{j - 7}{j + 6} \cdot \frac{j + 2}{j + 2} - \frac{2 j + 5}{(j + 2) (j + 6)}$

Step 3

Write each expression with a common denominator of $(j + 6) (j + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{j - 7}{j + 6}$ by $\frac{j + 2}{j + 2}$ .

$\frac{(j - 7) (j + 2)}{(j + 6) (j + 2)} - \frac{2 j + 5}{(j + 2) (j + 6)}$

Step 3.2

Reorder the factors of $(j + 6) (j + 2)$ .

$\frac{(j - 7) (j + 2)}{(j + 2) (j + 6)} - \frac{2 j + 5}{(j + 2) (j + 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(j - 7) (j + 2) - (2 j + 5)}{(j + 2) (j + 6)}$

Step 5

Rewrite $\frac{(j - 7) (j + 2) - (2 j + 5)}{(j + 2) (j + 6)}$ in a factored form.

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Step 5.1

Expand $(j - 7) (j + 2)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{j (j + 2) - 7 (j + 2) - (2 j + 5)}{(j + 2) (j + 6)}$

Step 5.1.2

Apply the distributive property.

$\frac{j \cdot j + j \cdot 2 - 7 (j + 2) - (2 j + 5)}{(j + 2) (j + 6)}$

Step 5.1.3

Apply the distributive property.

$\frac{j \cdot j + j \cdot 2 - 7 j - 7 \cdot 2 - (2 j + 5)}{(j + 2) (j + 6)}$

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $j$ by $j$ .

$\frac{j^{2} + j \cdot 2 - 7 j - 7 \cdot 2 - (2 j + 5)}{(j + 2) (j + 6)}$

Step 5.2.1.2

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

$\frac{j^{2} + 2 j - 7 j - 7 \cdot 2 - (2 j + 5)}{(j + 2) (j + 6)}$

Step 5.2.1.3

Multiply $- 7$ by $2$ .

$\frac{j^{2} + 2 j - 7 j - 14 - (2 j + 5)}{(j + 2) (j + 6)}$

Step 5.2.2

Subtract $7 j$ from $2 j$ .

$\frac{j^{2} - 5 j - 14 - (2 j + 5)}{(j + 2) (j + 6)}$

Step 5.3

Apply the distributive property.

$\frac{j^{2} - 5 j - 14 - (2 j) - 1 \cdot 5}{(j + 2) (j + 6)}$

Step 5.4

Multiply $2$ by $- 1$ .

$\frac{j^{2} - 5 j - 14 - 2 j - 1 \cdot 5}{(j + 2) (j + 6)}$

Step 5.5

Multiply $- 1$ by $5$ .

$\frac{j^{2} - 5 j - 14 - 2 j - 5}{(j + 2) (j + 6)}$

Step 5.6

Subtract $2 j$ from $- 5 j$ .

$\frac{j^{2} - 7 j - 14 - 5}{(j + 2) (j + 6)}$

Step 5.7

Subtract $5$ from $- 14$ .

$\frac{j^{2} - 7 j - 19}{(j + 2) (j + 6)}$