Basic Math Examples

$\frac{u}{u - 5} + \frac{u^{2} - 2}{5 - u}$

Step 1

Simplify terms.

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Step 1.1

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{u}{u - 5} + \frac{u^{2} - 2}{- 1 (- 5) - u}$

Step 1.2

Factor $- 1$ out of $- u$ .

$\frac{u}{u - 5} + \frac{u^{2} - 2}{- 1 (- 5) - (u)}$

Step 1.3

Factor $- 1$ out of $- 1 (- 5) - (u)$ .

$\frac{u}{u - 5} + \frac{u^{2} - 2}{- 1 (- 5 + u)}$

Step 1.4

Simplify the expression.

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Step 1.4.1

Move a negative from the denominator of $\frac{u^{2} - 2}{- 1 (- 5 + u)}$ to the numerator.

$\frac{u}{u - 5} + \frac{- (u^{2} - 2)}{- 5 + u}$

Step 1.4.2

Reorder terms.

$\frac{u}{u - 5} + \frac{- (u^{2} - 2)}{u - 5}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{u - (u^{2} - 2)}{u - 5}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the distributive property.

$\frac{u - u^{2} - - 2}{u - 5}$

Step 2.2

Multiply $- 1$ by $- 2$ .

$\frac{u - u^{2} + 2}{u - 5}$

Step 2.3

Reorder terms.

$\frac{- u^{2} + u + 2}{u - 5}$

Step 2.4

Factor by grouping.

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Step 2.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 2 = - 2$ and whose sum is $b = 1$ .

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Step 2.4.1.1

Multiply by $1$ .

$\frac{- u^{2} + 1 u + 2}{u - 5}$

Step 2.4.1.2

Rewrite $1$ as $- 1$ plus $2$

$\frac{- u^{2} + (- 1 + 2) u + 2}{u - 5}$

Step 2.4.1.3

Apply the distributive property.

$\frac{- u^{2} - 1 u + 2 u + 2}{u - 5}$

Step 2.4.2

Factor out the greatest common factor from each group.

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Step 2.4.2.1

Group the first two terms and the last two terms.

$\frac{(- u^{2} - 1 u) + 2 u + 2}{u - 5}$

Step 2.4.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{u (- u - 1) - 2 (- u - 1)}{u - 5}$

Step 2.4.3

Factor the polynomial by factoring out the greatest common factor, $- u - 1$ .

$\frac{(- u - 1) (u - 2)}{u - 5}$

Step 3

Simplify with factoring out.

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Step 3.1

Factor $- 1$ out of $- u$ .

$\frac{(- (u) - 1) (u - 2)}{u - 5}$

Step 3.2

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

$\frac{(- (u) - 1 (1)) (u - 2)}{u - 5}$

Step 3.3

Factor $- 1$ out of $- (u) - 1 (1)$ .

$\frac{- (u + 1) (u - 2)}{u - 5}$

Step 3.4

Simplify the expression.

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Step 3.4.1

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

$\frac{- 1 (u + 1) (u - 2)}{u - 5}$

Step 3.4.2

Move the negative in front of the fraction.

$- \frac{(u + 1) (u - 2)}{u - 5}$