Basic Math Examples

$\frac{y^{2} - 10 y + 24}{y^{2} - 3 y - 18} \div \frac{y^{2} + 2 y - 3}{y^{2} - 9 y + 8} \cdot (\frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24})$

Step 1

Factor $y^{2} - 10 y + 24$ 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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Step 1.2

Write the factored form using these integers.

$\frac{(y - 6) (y - 4)}{y^{2} - 3 y - 18} \div \frac{y^{2} + 2 y - 3}{y^{2} - 9 y + 8} \cdot \frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24}$

Step 2

Factor $y^{2} - 3 y - 18$ using the AC method.

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Step 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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(y - 6) (y - 4)}{(y - 6) (y + 3)} \div \frac{y^{2} + 2 y - 3}{y^{2} - 9 y + 8} \cdot \frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24}$

Step 3

Factor $y^{2} + 2 y - 3$ using the AC method.

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Step 3.1

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 $2$ .

$- 1, 3$

Step 3.2

Write the factored form using these integers.

$\frac{(y - 6) (y - 4)}{(y - 6) (y + 3)} \div \frac{(y - 1) (y + 3)}{y^{2} - 9 y + 8} \cdot \frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24}$

Step 4

Factor $y^{2} - 9 y + 8$ using the AC method.

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Step 4.1

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 $8$ and whose sum is $- 9$ .

$- 8, - 1$

Step 4.2

Write the factored form using these integers.

$\frac{(y - 6) (y - 4)}{(y - 6) (y + 3)} \div \frac{(y - 1) (y + 3)}{(y - 8) (y - 1)} \cdot \frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Reduce the expression $\frac{(y - 6) (y - 4)}{(y - 6) (y + 3)}$ by cancelling the common factors.

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Step 5.1.1

Cancel the common factor.

$\frac{(y - 6) (y - 4)}{(y - 6) (y + 3)} \div \frac{(y - 1) (y + 3)}{(y - 8) (y - 1)} \cdot \frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24}$

Step 5.1.2

Rewrite the expression.

$\frac{y - 4}{y + 3} \div \frac{(y - 1) (y + 3)}{(y - 8) (y - 1)} \cdot \frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24}$

Step 5.2

Reduce the expression $\frac{(y - 1) (y + 3)}{(y - 8) (y - 1)}$ by cancelling the common factors.

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Step 5.2.1

Cancel the common factor.

$\frac{y - 4}{y + 3} \div \frac{(y - 1) (y + 3)}{(y - 8) (y - 1)} \cdot \frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24}$

Step 5.2.2

Rewrite the expression.

$\frac{y - 4}{y + 3} \div \frac{y + 3}{y - 8} \cdot \frac{y^{2} + 6 y + 9}{y^{2} - 5 y - 24}$

Step 6

Factor using the perfect square rule.

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Step 6.1

Rewrite $9$ as $3^{2}$ .

$\frac{y - 4}{y + 3} \div \frac{y + 3}{y - 8} \cdot \frac{y^{2} + 6 y + 3^{2}}{y^{2} - 5 y - 24}$

Step 6.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 y = 2 \cdot y \cdot 3$

Step 6.3

Rewrite the polynomial.

$\frac{y - 4}{y + 3} \div \frac{y + 3}{y - 8} \cdot \frac{y^{2} + 2 \cdot y \cdot 3 + 3^{2}}{y^{2} - 5 y - 24}$

Step 6.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 3$ .

$\frac{y - 4}{y + 3} \div \frac{y + 3}{y - 8} \cdot \frac{{(y + 3)}^{2}}{y^{2} - 5 y - 24}$

Step 7

Factor $y^{2} - 5 y - 24$ using the AC method.

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Step 7.1

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 $- 24$ and whose sum is $- 5$ .

$- 8, 3$

Step 7.2

Write the factored form using these integers.

$\frac{y - 4}{y + 3} \div \frac{y + 3}{y - 8} \cdot \frac{{(y + 3)}^{2}}{(y - 8) (y + 3)}$

Step 8

Simplify terms.

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Step 8.1

Combine.

$\frac{\frac{y - 4}{y + 3} {(y + 3)}^{2}}{\frac{y + 3}{y - 8} ((y - 8) (y + 3))}$

Step 8.2

Cancel the common factor of ${(y + 3)}^{2}$ and $y + 3$ .

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Step 8.2.1

Factor $y + 3$ out of $\frac{y - 4}{y + 3} {(y + 3)}^{2}$ .

$\frac{(y + 3) (\frac{y - 4}{y + 3} (y + 3))}{\frac{y + 3}{y - 8} ((y - 8) (y + 3))}$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $y + 3$ out of $\frac{y + 3}{y - 8} ((y - 8) (y + 3))$ .

$\frac{(y + 3) (\frac{y - 4}{y + 3} (y + 3))}{(y + 3) (\frac{1}{y - 8} ((y - 8) (y + 3)))}$

Step 8.2.2.2

Cancel the common factor.

$\frac{(y + 3) (\frac{y - 4}{y + 3} (y + 3))}{(y + 3) (\frac{1}{y - 8} ((y - 8) (y + 3)))}$

Step 8.2.2.3

Rewrite the expression.

$\frac{\frac{y - 4}{y + 3} (y + 3)}{\frac{1}{y - 8} ((y - 8) (y + 3))}$

Step 8.3

Cancel the common factor of $y + 3$ .

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Step 8.3.1

Cancel the common factor.

$\frac{\frac{y - 4}{y + 3} (y + 3)}{\frac{1}{y - 8} ((y - 8) (y + 3))}$

Step 8.3.2

Rewrite the expression.

$\frac{\frac{y - 4}{y + 3}}{\frac{1}{y - 8} (y - 8)}$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$\frac{y - 4}{y + 3} \cdot \frac{1}{\frac{1}{y - 8} (y - 8)}$

Step 10

Factor $\frac{1}{y - 8}$ out of $\frac{1}{\frac{1}{y - 8} (y - 8)}$ .

$\frac{y - 4}{y + 3} ((y - 8) \frac{1}{y - 8})$

Step 11

Cancel the common factor of $y - 8$ .

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Step 11.1

Cancel the common factor.

$\frac{y - 4}{y + 3} ((y - 8) \frac{1}{y - 8})$

Step 11.2

Rewrite the expression.

$\frac{y - 4}{y + 3} \cdot 1$

Step 12

Multiply $\frac{y - 4}{y + 3}$ by $1$ .

$\frac{y - 4}{y + 3}$