Basic Math Examples

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

Step 1

Factor $y^{2} - 7 y + 6$ 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 $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 1.2

Write the factored form using these integers.

$\frac{(y - 6) (y - 1)}{- 4 y^{2} + 8 y - 4}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $4$ out of $- 4 y^{2} + 8 y - 4$ .

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Step 2.1.1

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

$\frac{(y - 6) (y - 1)}{4 (- y^{2}) + 8 y - 4}$

Step 2.1.2

Factor $4$ out of $8 y$ .

$\frac{(y - 6) (y - 1)}{4 (- y^{2}) + 4 (2 y) - 4}$

Step 2.1.3

Factor $4$ out of $- 4$ .

$\frac{(y - 6) (y - 1)}{4 (- y^{2}) + 4 (2 y) + 4 (- 1)}$

Step 2.1.4

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

$\frac{(y - 6) (y - 1)}{4 (- y^{2} + 2 y) + 4 (- 1)}$

Step 2.1.5

Factor $4$ out of $4 (- y^{2} + 2 y) + 4 (- 1)$ .

$\frac{(y - 6) (y - 1)}{4 (- y^{2} + 2 y - 1)}$

Step 2.2

Factor by grouping.

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Step 2.2.1

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 $a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is $b = 2$ .

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Step 2.2.1.1

Factor $2$ out of $2 y$ .

$\frac{(y - 6) (y - 1)}{4 (- y^{2} + 2 (y) - 1)}$

Step 2.2.1.2

Rewrite $2$ as $1$ plus $1$

$\frac{(y - 6) (y - 1)}{4 (- y^{2} + (1 + 1) y - 1)}$

Step 2.2.1.3

Apply the distributive property.

$\frac{(y - 6) (y - 1)}{4 (- y^{2} + 1 y + 1 y - 1)}$

Step 2.2.1.4

Multiply $y$ by $1$ .

$\frac{(y - 6) (y - 1)}{4 (- y^{2} + y + 1 y - 1)}$

Step 2.2.1.5

Multiply $y$ by $1$ .

$\frac{(y - 6) (y - 1)}{4 (- y^{2} + y + y - 1)}$

Step 2.2.2

Factor out the greatest common factor from each group.

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Step 2.2.2.1

Group the first two terms and the last two terms.

$\frac{(y - 6) (y - 1)}{4 ((- y^{2} + y) + y - 1)}$

Step 2.2.2.2

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

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

Step 2.2.3

Factor the polynomial by factoring out the greatest common factor, $- y + 1$ .

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

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

Step 2.3

Combine exponents.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Raise $y - 1$ to the power of $1$ .

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

Step 2.3.6

Raise $y - 1$ to the power of $1$ .

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

Step 2.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.8

Add $1$ and $1$ .

$\frac{(y - 6) (y - 1)}{4 \cdot - 1 {(y - 1)}^{2}}$

Step 2.3.9

Multiply $4$ by $- 1$ .

$\frac{(y - 6) (y - 1)}{- 4 {(y - 1)}^{2}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $y - 1$ and ${(y - 1)}^{2}$ .

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Step 3.1.1

Factor $y - 1$ out of $(y - 6) (y - 1)$ .

$\frac{(y - 1) (y - 6)}{- 4 {(y - 1)}^{2}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $y - 1$ out of $- 4 {(y - 1)}^{2}$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

Move the negative in front of the fraction.

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