Algebra Examples

$\frac{b^{2} - 4 b y}{2 y^{2} - b y} - \frac{4 y}{b - 2 y}$

Step 1

Simplify each term.

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Step 1.1

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

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Step 1.1.1

Factor $b$ out of $b^{2}$ .

$\frac{b \cdot b - 4 b y}{2 y^{2} - b y} - \frac{4 y}{b - 2 y}$

Step 1.1.2

Factor $b$ out of $- 4 b y$ .

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

Step 1.1.3

Factor $b$ out of $b \cdot b + b (- 4 y)$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Factor $y$ out of $- b y$ .

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

Step 1.2.3

Factor $y$ out of $y (2 y) + y (- b)$ .

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

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $b$ .

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

Step 2.2

Factor $- 1$ out of $- 2 y$ .

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

Step 2.3

Factor $- 1$ out of $- 1 (- b) - (2 y)$ .

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

Step 2.4

Reorder terms.

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

Step 3

To write $\frac{b (b - 4 y)}{y (- b + 2 y)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{b (b - 4 y)}{y (- b + 2 y)} \cdot \frac{- 1}{- 1} - \frac{4 y}{- 1 (- b + 2 y)}$

Step 4

To write $- \frac{4 y}{- 1 (- b + 2 y)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{b (b - 4 y)}{y (- b + 2 y)} \cdot \frac{- 1}{- 1} - \frac{4 y}{- 1 (- b + 2 y)} \cdot \frac{y}{y}$

Step 5

Write each expression with a common denominator of $y (- b + 2 y) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{b (b - 4 y)}{y (- b + 2 y)}$ by $\frac{- 1}{- 1}$ .

$\frac{b (b - 4 y) \cdot - 1}{y (- b + 2 y) \cdot - 1} - \frac{4 y}{- 1 (- b + 2 y)} \cdot \frac{y}{y}$

Step 5.2

Multiply $\frac{4 y}{- 1 (- b + 2 y)}$ by $\frac{y}{y}$ .

$\frac{b (b - 4 y) \cdot - 1}{y (- b + 2 y) \cdot - 1} - \frac{4 y \cdot y}{- 1 (- b + 2 y) y}$

Step 5.3

Reorder the factors of $y (- b + 2 y) \cdot - 1$ .

$\frac{b (b - 4 y) \cdot - 1}{- y (- b + 2 y)} - \frac{4 y \cdot y}{- 1 (- b + 2 y) y}$

Step 5.4

Reorder the factors of $- 1 (- b + 2 y) y$ .

$\frac{b (b - 4 y) \cdot - 1}{- y (- b + 2 y)} - \frac{4 y \cdot y}{- y (- b + 2 y)}$

Step 6

Combine the numerators over the common denominator.

$\frac{b (b - 4 y) \cdot - 1 - 4 y \cdot y}{- y (- b + 2 y)}$

Step 7

Simplify the numerator.

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Step 7.1

Apply the distributive property.

$\frac{(b \cdot b + b (- 4 y)) \cdot - 1 - 4 y \cdot y}{- y (- b + 2 y)}$

Step 7.2

Multiply $b$ by $b$ .

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

Step 7.3

Rewrite using the commutative property of multiplication.

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

Step 7.4

Apply the distributive property.

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

Step 7.5

Move $- 1$ to the left of $b^{2}$ .

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

Step 7.6

Multiply $- 1$ by $- 4$ .

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

Step 7.7

Rewrite $- 1 b^{2}$ as $- b^{2}$ .

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

Step 7.8

Multiply $y$ by $y$ by adding the exponents.

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Step 7.8.1

Move $y$ .

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

Step 7.8.2

Multiply $y$ by $y$ .

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

Step 7.9

Factor by grouping.

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Step 7.9.1

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 $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = 4$ .

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Step 7.9.1.1

Reorder terms.

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

Step 7.9.1.2

Reorder $- 4 y^{2}$ and $4 b y$ .

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

Step 7.9.1.3

Factor $4$ out of $4 b y$ .

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

Step 7.9.1.4

Rewrite $4$ as $2$ plus $2$

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

Step 7.9.1.5

Apply the distributive property.

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

Step 7.9.1.6

Move parentheses.

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

Step 7.9.2

Factor out the greatest common factor from each group.

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Step 7.9.2.1

Group the first two terms and the last two terms.

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

Step 7.9.2.2

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

$\frac{b (- b + 2 y) - 2 y (- b + 2 y)}{- y (- b + 2 y)}$

Step 7.9.3

Factor the polynomial by factoring out the greatest common factor, $- b + 2 y$ .

$\frac{(- b + 2 y) (b - 2 y)}{- y (- b + 2 y)}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $- 1$ out of $- b$ .

$\frac{(- (b) + 2 y) (b - 2 y)}{- y (- b + 2 y)}$

Step 8.2

Factor $- 1$ out of $2 y$ .

$\frac{(- (b) - (- 2 y)) (b - 2 y)}{- y (- b + 2 y)}$

Step 8.3

Factor $- 1$ out of $- (b) - (- 2 y)$ .

$\frac{- (b - 2 y) (b - 2 y)}{- y (- b + 2 y)}$

Step 8.4

Rewrite $- (b - 2 y)$ as $- 1 (b - 2 y)$ .

$\frac{- 1 (b - 2 y) (b - 2 y)}{- y (- b + 2 y)}$

Step 8.5

Raise $b - 2 y$ to the power of $1$ .

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

Step 8.6

Raise $b - 2 y$ to the power of $1$ .

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

Step 8.7

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

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

Step 8.8

Add $1$ and $1$ .

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

Step 9

Simplify terms.

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Step 9.1

Dividing two negative values results in a positive value.

$\frac{{(b - 2 y)}^{2}}{y (- b + 2 y)}$

Step 9.2

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

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Step 9.2.1

Factor $- 1$ out of $b$ .

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

Step 9.2.2

Factor $- 1$ out of $- 2 y$ .

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

Step 9.2.3

Factor $- 1$ out of $- 1 (- b) - (2 y)$ .

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

Step 9.2.4

Apply the product rule to $- 1 (- b + 2 y)$ .

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

Step 9.2.5

Raise $- 1$ to the power of $2$ .

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

Step 9.2.6

Multiply ${(- b + 2 y)}^{2}$ by $1$ .

$\frac{{(- b + 2 y)}^{2}}{y (- b + 2 y)}$

Step 9.2.7

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

$\frac{(- b + 2 y) (- b + 2 y)}{y (- b + 2 y)}$

Step 9.2.8

Cancel the common factors.

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Step 9.2.8.1

Factor $- b + 2 y$ out of $y (- b + 2 y)$ .

$\frac{(- b + 2 y) (- b + 2 y)}{(- b + 2 y) y}$

Step 9.2.8.2

Cancel the common factor.

$\frac{(- b + 2 y) (- b + 2 y)}{(- b + 2 y) y}$

Step 9.2.8.3

Rewrite the expression.

$\frac{- b + 2 y}{y}$

Step 9.3

Factor $- 1$ out of $- b$ .

$\frac{- (b) + 2 y}{y}$

Step 9.4

Factor $- 1$ out of $2 y$ .

$\frac{- (b) - (- 2 y)}{y}$

Step 9.5

Factor $- 1$ out of $- (b) - (- 2 y)$ .

$\frac{- (b - 2 y)}{y}$

Step 9.6

Simplify the expression.

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Step 9.6.1

Rewrite $- (b - 2 y)$ as $- 1 (b - 2 y)$ .

$\frac{- 1 (b - 2 y)}{y}$

Step 9.6.2

Move the negative in front of the fraction.

$- \frac{b - 2 y}{y}$