Basic Math Examples

$\frac{y^{2} - 10 y + 24}{96 - 6 y^{2}}$

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.

$\frac{(y - 6) (y - 4)}{96 - 6 y^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $6$ out of $96 - 6 y^{2}$ .

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Step 2.1.1

Factor $6$ out of $96$ .

$\frac{(y - 6) (y - 4)}{6 (16) - 6 y^{2}}$

Step 2.1.2

Factor $6$ out of $- 6 y^{2}$ .

$\frac{(y - 6) (y - 4)}{6 (16) + 6 (- y^{2})}$

Step 2.1.3

Factor $6$ out of $6 (16) + 6 (- y^{2})$ .

$\frac{(y - 6) (y - 4)}{6 (16 - y^{2})}$

Step 2.2

Rewrite $16$ as $4^{2}$ .

$\frac{(y - 6) (y - 4)}{6 (4^{2} - y^{2})}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = y$ .

$\frac{(y - 6) (y - 4)}{6 (4 + y) (4 - y)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $y - 4$ and $4 - y$ .

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Step 3.1.1

Factor $- 1$ out of $y$ .

$\frac{(y - 6) (- 1 (- y) - 4)}{6 (4 + y) (4 - y)}$

Step 3.1.2

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{(y - 6) (- 1 (- y) - 1 (4))}{6 (4 + y) (4 - y)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- y) - 1 (4)$ .

$\frac{(y - 6) (- 1 (- y + 4))}{6 (4 + y) (4 - y)}$

Step 3.1.4

Reorder terms.

$\frac{(y - 6) (- 1 (- y + 4))}{6 (4 + y) (- y + 4)}$

Step 3.1.5

Cancel the common factor.

$\frac{(y - 6) (- 1 (- y + 4))}{6 (4 + y) (- y + 4)}$

Step 3.1.6

Rewrite the expression.

$\frac{(y - 6) \cdot (- 1)}{6 (4 + y)}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move $- 1$ to the left of $y - 6$ .

$\frac{- 1 \cdot (y - 6)}{6 (4 + y)}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{y - 6}{6 (4 + y)}$