Finite Math Examples

$\frac{4 a}{a - 2} - \frac{3 a}{a - 3} + \frac{4 a}{a^{2} - 5 a + 6}$

Step 1

Factor $a^{2} - 5 a + 6$ 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.

$\frac{4 a}{a - 2} - \frac{3 a}{a - 3} + \frac{4 a}{(a - 3) (a - 2)}$

Step 2

To write $\frac{4 a}{a - 2}$ as a fraction with a common denominator, multiply by $\frac{a - 3}{a - 3}$ .

$\frac{4 a}{a - 2} \cdot \frac{a - 3}{a - 3} - \frac{3 a}{a - 3} + \frac{4 a}{(a - 3) (a - 2)}$

Step 3

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

$\frac{4 a}{a - 2} \cdot \frac{a - 3}{a - 3} - \frac{3 a}{a - 3} \cdot \frac{a - 2}{a - 2} + \frac{4 a}{(a - 3) (a - 2)}$

Step 4

Write each expression with a common denominator of $(a - 2) (a - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{4 a}{a - 2}$ by $\frac{a - 3}{a - 3}$ .

$\frac{4 a (a - 3)}{(a - 2) (a - 3)} - \frac{3 a}{a - 3} \cdot \frac{a - 2}{a - 2} + \frac{4 a}{(a - 3) (a - 2)}$

Step 4.2

Multiply $\frac{3 a}{a - 3}$ by $\frac{a - 2}{a - 2}$ .

$\frac{4 a (a - 3)}{(a - 2) (a - 3)} - \frac{3 a (a - 2)}{(a - 3) (a - 2)} + \frac{4 a}{(a - 3) (a - 2)}$

Step 4.3

Reorder the factors of $(a - 3) (a - 2)$ .

$\frac{4 a (a - 3)}{(a - 2) (a - 3)} - \frac{3 a (a - 2)}{(a - 2) (a - 3)} + \frac{4 a}{(a - 3) (a - 2)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{4 a (a - 3) - 3 a (a - 2)}{(a - 2) (a - 3)} + \frac{4 a}{(a - 3) (a - 2)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{4 a (a - 3) - 3 a (a - 2) + 4 a}{(a - 2) (a - 3)}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{4 a \cdot a + 4 a \cdot - 3 - 3 a (a - 2) + 4 a}{(a - 2) (a - 3)}$

Step 6.2

Multiply $a$ by $a$ by adding the exponents.

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Step 6.2.1

Move $a$ .

$\frac{4 (a \cdot a) + 4 a \cdot - 3 - 3 a (a - 2) + 4 a}{(a - 2) (a - 3)}$

Step 6.2.2

Multiply $a$ by $a$ .

$\frac{4 a^{2} + 4 a \cdot - 3 - 3 a (a - 2) + 4 a}{(a - 2) (a - 3)}$

Step 6.3

Multiply $- 3$ by $4$ .

$\frac{4 a^{2} - 12 a - 3 a (a - 2) + 4 a}{(a - 2) (a - 3)}$

Step 6.4

Apply the distributive property.

$\frac{4 a^{2} - 12 a - 3 a \cdot a - 3 a \cdot - 2 + 4 a}{(a - 2) (a - 3)}$

Step 6.5

Multiply $a$ by $a$ by adding the exponents.

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Step 6.5.1

Move $a$ .

$\frac{4 a^{2} - 12 a - 3 (a \cdot a) - 3 a \cdot - 2 + 4 a}{(a - 2) (a - 3)}$

Step 6.5.2

Multiply $a$ by $a$ .

$\frac{4 a^{2} - 12 a - 3 a^{2} - 3 a \cdot - 2 + 4 a}{(a - 2) (a - 3)}$

Step 6.6

Multiply $- 2$ by $- 3$ .

$\frac{4 a^{2} - 12 a - 3 a^{2} + 6 a + 4 a}{(a - 2) (a - 3)}$

Step 7

Simplify terms.

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Step 7.1

Subtract $3 a^{2}$ from $4 a^{2}$ .

$\frac{a^{2} - 12 a + 6 a + 4 a}{(a - 2) (a - 3)}$

Step 7.2

Add $- 12 a$ and $6 a$ .

$\frac{a^{2} - 6 a + 4 a}{(a - 2) (a - 3)}$

Step 7.3

Add $- 6 a$ and $4 a$ .

$\frac{a^{2} - 2 a}{(a - 2) (a - 3)}$

Step 7.4

Factor $a$ out of $a^{2} - 2 a$ .

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Step 7.4.1

Factor $a$ out of $a^{2}$ .

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

Step 7.4.2

Factor $a$ out of $- 2 a$ .

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

Step 7.4.3

Factor $a$ out of $a \cdot a + a \cdot - 2$ .

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

Step 7.5

Cancel the common factor of $a - 2$ .

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Step 7.5.1

Cancel the common factor.

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

Step 7.5.2

Rewrite the expression.

$\frac{a}{a - 3}$