Basic Math Examples

$\frac{- 4 y^{2} + 28 y - 48}{y^{2} - 6 y + 8}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4$ out of $- 4 y^{2} + 28 y - 48$ .

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Step 1.1.1

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

$\frac{4 (- y^{2}) + 28 y - 48}{y^{2} - 6 y + 8}$

Step 1.1.2

Factor $4$ out of $28 y$ .

$\frac{4 (- y^{2}) + 4 (7 y) - 48}{y^{2} - 6 y + 8}$

Step 1.1.3

Factor $4$ out of $- 48$ .

$\frac{4 (- y^{2}) + 4 (7 y) + 4 (- 12)}{y^{2} - 6 y + 8}$

Step 1.1.4

Factor $4$ out of $4 (- y^{2}) + 4 (7 y)$ .

$\frac{4 (- y^{2} + 7 y) + 4 (- 12)}{y^{2} - 6 y + 8}$

Step 1.1.5

Factor $4$ out of $4 (- y^{2} + 7 y) + 4 (- 12)$ .

$\frac{4 (- y^{2} + 7 y - 12)}{y^{2} - 6 y + 8}$

Step 1.2

Factor by grouping.

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Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 12 = 12$ and whose sum is $b = 7$ .

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Step 1.2.1.1

Factor $7$ out of $7 y$ .

$\frac{4 (- y^{2} + 7 (y) - 12)}{y^{2} - 6 y + 8}$

Step 1.2.1.2

Rewrite $7$ as $3$ plus $4$

$\frac{4 (- y^{2} + (3 + 4) y - 12)}{y^{2} - 6 y + 8}$

Step 1.2.1.3

Apply the distributive property.

$\frac{4 (- y^{2} + 3 y + 4 y - 12)}{y^{2} - 6 y + 8}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

$\frac{4 ((- y^{2} + 3 y) + 4 y - 12)}{y^{2} - 6 y + 8}$

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{4 (y (- y + 3) - 4 (- y + 3))}{y^{2} - 6 y + 8}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $- y + 3$ .

$\frac{4 ((- y + 3) (y - 4))}{y^{2} - 6 y + 8}$

$\frac{4 (- y + 3) (y - 4)}{y^{2} - 6 y + 8}$

Step 2

Factor $y^{2} - 6 y + 8$ 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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $y - 4$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Factor $- 1$ out of $- y$ .

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

Step 3.3

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{4 (- (y) - 1 (- 3))}{y - 2}$

Step 3.4

Factor $- 1$ out of $- (y) - 1 (- 3)$ .

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

Step 3.5

Simplify the expression.

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Step 3.5.1

Rewrite $- (y - 3)$ as $- 1 (y - 3)$ .

$\frac{4 (- 1 (y - 3))}{y - 2}$

Step 3.5.2

Move the negative in front of the fraction.

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