Basic Math Examples

$\frac{h}{h^{2} - 5 h + 6} + \frac{3}{3 - h}$

Step 1

Factor $h^{2} - 5 h + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 1.2

Write the factored form using these integers.

$\frac{h}{(h - 3) (h - 2)} + \frac{3}{3 - h}$

Step 2

Simplify with factoring out.

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Step 2.1

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 2.2

Factor $- 1$ out of $- h$ .

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

Step 2.3

Factor $- 1$ out of $- 1 (- 3) - (h)$ .

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{3}{- 1 (- 3 + h)}$ to the numerator.

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

Step 2.4.2

Reorder terms.

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

Step 3

To write $\frac{- 1 \cdot 3}{h - 3}$ as a fraction with a common denominator, multiply by $\frac{h - 2}{h - 2}$ .

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

Step 4

Simplify terms.

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Step 4.1

Multiply $\frac{- 1 \cdot 3}{h - 3}$ by $\frac{h - 2}{h - 2}$ .

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $3$ .

$\frac{h - 3 (h - 2)}{(h - 3) (h - 2)}$

Step 5.2

Apply the distributive property.

$\frac{h - 3 h - 3 \cdot - 2}{(h - 3) (h - 2)}$

Step 5.3

Multiply $- 3$ by $- 2$ .

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

Step 5.4

Subtract $3 h$ from $h$ .

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

Step 5.5

Factor $2$ out of $- 2 h + 6$ .

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Step 5.5.1

Factor $2$ out of $- 2 h$ .

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

Step 5.5.2

Factor $2$ out of $6$ .

$\frac{2 (- h) + 2 (3)}{(h - 3) (h - 2)}$

Step 5.5.3

Factor $2$ out of $2 (- h) + 2 (3)$ .

$\frac{2 (- h + 3)}{(h - 3) (h - 2)}$

Step 6

Cancel the common factor of $- h + 3$ and $h - 3$ .

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Step 6.1

Factor $- 1$ out of $- h$ .

$\frac{2 (- (h) + 3)}{(h - 3) (h - 2)}$

Step 6.2

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{2 (- (h) - 1 (- 3))}{(h - 3) (h - 2)}$

Step 6.3

Factor $- 1$ out of $- (h) - 1 (- 3)$ .

$\frac{2 (- (h - 3))}{(h - 3) (h - 2)}$

Step 6.4

Rewrite $- (h - 3)$ as $- 1 (h - 3)$ .

$\frac{2 (- 1 (h - 3))}{(h - 3) (h - 2)}$

Step 6.5

Cancel the common factor.

$\frac{2 (- 1 (h - 3))}{(h - 3) (h - 2)}$

Step 6.6

Rewrite the expression.

$\frac{2 \cdot (- 1)}{h - 2}$

Step 7

Multiply $2$ by $- 1$ .

$\frac{- 2}{h - 2}$

Step 8

Move the negative in front of the fraction.

$- \frac{2}{h - 2}$