Basic Math Examples

$\frac{u - 1}{u + 1} - \frac{u^{2} - u - 6}{u^{2} + 4 u + 3}$

Step 1

Simplify each term.

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Step 1.1

Factor $u^{2} - u - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{u - 1}{u + 1} - \frac{(u - 3) (u + 2)}{u^{2} + 4 u + 3}$

Step 1.2

Factor $u^{2} + 4 u + 3$ 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 $3$ and whose sum is $4$ .

$1, 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{u - 1}{u + 1} - \frac{(u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 2

To write $\frac{u - 1}{u + 1}$ as a fraction with a common denominator, multiply by $\frac{u + 3}{u + 3}$ .

$\frac{u - 1}{u + 1} \cdot \frac{u + 3}{u + 3} - \frac{(u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{u - 1}{u + 1}$ by $\frac{u + 3}{u + 3}$ .

$\frac{(u - 1) (u + 3)}{(u + 1) (u + 3)} - \frac{(u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{(u - 1) (u + 3) - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(u - 1) (u + 3)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{u (u + 3) - 1 (u + 3) - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.1.2

Apply the distributive property.

$\frac{u \cdot u + u \cdot 3 - 1 (u + 3) - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.1.3

Apply the distributive property.

$\frac{u \cdot u + u \cdot 3 - 1 u - 1 \cdot 3 - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $u$ by $u$ .

$\frac{u^{2} + u \cdot 3 - 1 u - 1 \cdot 3 - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.2.1.2

Move $3$ to the left of $u$ .

$\frac{u^{2} + 3 \cdot u - 1 u - 1 \cdot 3 - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.2.1.3

Rewrite $- 1 u$ as $- u$ .

$\frac{u^{2} + 3 u - u - 1 \cdot 3 - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.2.1.4

Multiply $- 1$ by $3$ .

$\frac{u^{2} + 3 u - u - 3 - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.2.2

Subtract $u$ from $3 u$ .

$\frac{u^{2} + 2 u - 3 - (u - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.3

Apply the distributive property.

$\frac{u^{2} + 2 u - 3 + (- u - - 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.4

Multiply $- 1$ by $- 3$ .

$\frac{u^{2} + 2 u - 3 + (- u + 3) (u + 2)}{(u + 1) (u + 3)}$

Step 4.5

Expand $(- u + 3) (u + 2)$ using the FOIL Method.

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Step 4.5.1

Apply the distributive property.

$\frac{u^{2} + 2 u - 3 - u (u + 2) + 3 (u + 2)}{(u + 1) (u + 3)}$

Step 4.5.2

Apply the distributive property.

$\frac{u^{2} + 2 u - 3 - u \cdot u - u \cdot 2 + 3 (u + 2)}{(u + 1) (u + 3)}$

Step 4.5.3

Apply the distributive property.

$\frac{u^{2} + 2 u - 3 - u \cdot u - u \cdot 2 + 3 u + 3 \cdot 2}{(u + 1) (u + 3)}$

Step 4.6

Simplify and combine like terms.

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Step 4.6.1

Simplify each term.

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Step 4.6.1.1

Multiply $u$ by $u$ by adding the exponents.

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Step 4.6.1.1.1

Move $u$ .

$\frac{u^{2} + 2 u - 3 - (u \cdot u) - u \cdot 2 + 3 u + 3 \cdot 2}{(u + 1) (u + 3)}$

Step 4.6.1.1.2

Multiply $u$ by $u$ .

$\frac{u^{2} + 2 u - 3 - u^{2} - u \cdot 2 + 3 u + 3 \cdot 2}{(u + 1) (u + 3)}$

Step 4.6.1.2

Multiply $2$ by $- 1$ .

$\frac{u^{2} + 2 u - 3 - u^{2} - 2 u + 3 u + 3 \cdot 2}{(u + 1) (u + 3)}$

Step 4.6.1.3

Multiply $3$ by $2$ .

$\frac{u^{2} + 2 u - 3 - u^{2} - 2 u + 3 u + 6}{(u + 1) (u + 3)}$

Step 4.6.2

Add $- 2 u$ and $3 u$ .

$\frac{u^{2} + 2 u - 3 - u^{2} + u + 6}{(u + 1) (u + 3)}$

Step 4.7

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

$\frac{2 u - 3 + 0 + u + 6}{(u + 1) (u + 3)}$

Step 4.8

Add $2 u$ and $0$ .

$\frac{2 u - 3 + u + 6}{(u + 1) (u + 3)}$

Step 4.9

Add $2 u$ and $u$ .

$\frac{3 u - 3 + 6}{(u + 1) (u + 3)}$

Step 4.10

Add $- 3$ and $6$ .

$\frac{3 u + 3}{(u + 1) (u + 3)}$

Step 4.11

Factor $3$ out of $3 u + 3$ .

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Step 4.11.1

Factor $3$ out of $3 u$ .

$\frac{3 (u) + 3}{(u + 1) (u + 3)}$

Step 4.11.2

Factor $3$ out of $3$ .

$\frac{3 (u) + 3 (1)}{(u + 1) (u + 3)}$

Step 4.11.3

Factor $3$ out of $3 (u) + 3 (1)$ .

$\frac{3 (u + 1)}{(u + 1) (u + 3)}$

Step 5

Cancel the common factor of $u + 1$ .

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Step 5.1

Cancel the common factor.

$\frac{3 (u + 1)}{(u + 1) (u + 3)}$

Step 5.2

Rewrite the expression.

$\frac{3}{u + 3}$