Finite Math Examples

$\frac{(a^{2} + 3 a - 54) \cdot (a^{2} - 9 a - 10)}{(a^{2} + 4 a - 60) \cdot (a^{2} + 10 a + 9)}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $a^{2} + 3 a - 54$ 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 $- 54$ and whose sum is $3$ .

$- 6, 9$

Step 1.1.2

Write the factored form using these integers.

$\frac{(a - 6) (a + 9) (a^{2} - 9 a - 10)}{(a^{2} + 4 a - 60) \cdot (a^{2} + 10 a + 9)}$

Step 1.2

Factor $a^{2} - 9 a - 10$ 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 $- 10$ and whose sum is $- 9$ .

$- 10, 1$

Step 1.2.2

Write the factored form using these integers.

$\frac{(a - 6) (a + 9) ((a - 10) (a + 1))}{(a^{2} + 4 a - 60) \cdot (a^{2} + 10 a + 9)}$

$\frac{(a - 6) (a + 9) (a - 10) (a + 1)}{(a^{2} + 4 a - 60) \cdot (a^{2} + 10 a + 9)}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $a^{2} + 4 a - 60$ using the AC method.

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Step 2.1.1

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 $- 60$ and whose sum is $4$ .

$- 6, 10$

Step 2.1.2

Write the factored form using these integers.

$\frac{(a - 6) (a + 9) (a - 10) (a + 1)}{(a - 6) (a + 10) (a^{2} + 10 a + 9)}$

Step 2.2

Factor $a^{2} + 10 a + 9$ using the AC method.

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Step 2.2.1

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 $9$ and whose sum is $10$ .

$1, 9$

Step 2.2.2

Write the factored form using these integers.

$\frac{(a - 6) (a + 9) (a - 10) (a + 1)}{(a - 6) (a + 10) ((a + 1) (a + 9))}$

$\frac{(a - 6) (a + 9) (a - 10) (a + 1)}{(a - 6) (a + 10) (a + 1) (a + 9)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $a - 6$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(a - 6) (a + 9) (a - 10) (a + 1)}{(a - 6) (a + 10) (a + 1) (a + 9)}$

Step 3.1.2

Rewrite the expression.

$\frac{((a + 9) (a - 10)) (a + 1)}{((a + 10) (a + 1)) (a + 9)}$

Step 3.2

Cancel the common factor of $a + 9$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(a + 9) (a - 10) (a + 1)}{(a + 10) (a + 1) (a + 9)}$

Step 3.2.2

Rewrite the expression.

$\frac{(a - 10) (a + 1)}{(a + 10) (a + 1)}$

Step 3.3

Cancel the common factor of $a + 1$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(a - 10) (a + 1)}{(a + 10) (a + 1)}$

Step 3.3.2

Rewrite the expression.

$\frac{a - 10}{a + 10}$