Algebra Examples

$\frac{\frac{y^{2} - 5 y + 6}{y^{3}}}{\frac{y^{2} + 3 y - 10}{4 y^{2}}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - 5 y + 6}{y^{3}} \cdot \frac{4 y^{2}}{y^{2} + 3 y - 10}$

Step 2

Cancel the common factor of $y^{2}$ .

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Step 2.1

Factor $y^{2}$ out of $y^{3}$ .

$\frac{y^{2} - 5 y + 6}{y^{2} y} \cdot \frac{4 y^{2}}{y^{2} + 3 y - 10}$

Step 2.2

Factor $y^{2}$ out of $4 y^{2}$ .

$\frac{y^{2} - 5 y + 6}{y^{2} y} \cdot \frac{y^{2} \cdot 4}{y^{2} + 3 y - 10}$

Step 2.3

Cancel the common factor.

$\frac{y^{2} - 5 y + 6}{y^{2} y} \cdot \frac{y^{2} \cdot 4}{y^{2} + 3 y - 10}$

Step 2.4

Rewrite the expression.

$\frac{y^{2} - 5 y + 6}{y} \cdot \frac{4}{y^{2} + 3 y - 10}$

Step 3

Multiply $\frac{y^{2} - 5 y + 6}{y}$ by $\frac{4}{y^{2} + 3 y - 10}$ .

$\frac{(y^{2} - 5 y + 6) \cdot 4}{y (y^{2} + 3 y - 10)}$

Step 4

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

$- 3, - 2$

Step 4.2

Write the factored form using these integers.

$\frac{(y - 3) (y - 2) \cdot 4}{y (y^{2} + 3 y - 10)}$

Step 5

Factor $y^{2} + 3 y - 10$ using the AC method.

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Step 5.1

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 $3$ .

$- 2, 5$

Step 5.2

Write the factored form using these integers.

$\frac{(y - 3) (y - 2) \cdot 4}{y ((y - 2) (y + 5))}$

$\frac{(y - 3) (y - 2) \cdot 4}{y (y - 2) (y + 5)}$

Step 6

Cancel the common factor of $y - 2$ .

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Step 6.1

Cancel the common factor.

$\frac{(y - 3) (y - 2) \cdot 4}{y (y - 2) (y + 5)}$

Step 6.2

Rewrite the expression.

$\frac{(y - 3) \cdot 4}{y (y + 5)}$

Step 7

Move $4$ to the left of $y - 3$ .

$\frac{4 (y - 3)}{y (y + 5)}$

Step 8

Apply the distributive property.

$\frac{4 y + 4 \cdot - 3}{y (y + 5)}$

Step 9

Multiply $4$ by $- 3$ .

$\frac{4 y - 12}{y (y + 5)}$

Step 10

Split the fraction $\frac{4 y - 12}{y (y + 5)}$ into two fractions.

$\frac{4 y}{y (y + 5)} + \frac{- 12}{y (y + 5)}$

Step 11

Cancel the common factor of $y$ .

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Step 11.1

Cancel the common factor.

$\frac{4 y}{y (y + 5)} + \frac{- 12}{y (y + 5)}$

Step 11.2

Rewrite the expression.

$\frac{4}{y + 5} + \frac{- 12}{y (y + 5)}$

Step 12

Move the negative in front of the fraction.

$\frac{4}{y + 5} - \frac{12}{y (y + 5)}$