Algebra Examples

$\frac{1}{4} x^{2} - \frac{3}{8} x y + \frac{1}{2} y^{2} - \frac{1}{2} x y + \frac{1}{4} y^{2} - \frac{3}{8} x^{2}$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{1}{4}$ and $x^{2}$ .

$\frac{x^{2}}{4} - \frac{3}{8} \cdot (x y) + \frac{1}{2} y^{2} - \frac{1}{2} \cdot (x y) + \frac{1}{4} y^{2} - \frac{3}{8} x^{2}$

Step 1.2

Multiply $- \frac{3}{8} (x y)$ .

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Step 1.2.1

Combine $x$ and $\frac{3}{8}$ .

$\frac{x^{2}}{4} - \frac{x \cdot 3}{8} y + \frac{1}{2} y^{2} - \frac{1}{2} \cdot (x y) + \frac{1}{4} y^{2} - \frac{3}{8} x^{2}$

Step 1.2.2

Combine $y$ and $\frac{x \cdot 3}{8}$ .

$\frac{x^{2}}{4} - \frac{y (x \cdot 3)}{8} + \frac{1}{2} y^{2} - \frac{1}{2} \cdot (x y) + \frac{1}{4} y^{2} - \frac{3}{8} x^{2}$

Step 1.3

Move $3$ to the left of $y x$ .

$\frac{x^{2}}{4} - \frac{3 \cdot (y x)}{8} + \frac{1}{2} y^{2} - \frac{1}{2} \cdot (x y) + \frac{1}{4} y^{2} - \frac{3}{8} x^{2}$

Step 1.4

Combine $\frac{1}{2}$ and $y^{2}$ .

$\frac{x^{2}}{4} - \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{1}{2} \cdot (x y) + \frac{1}{4} y^{2} - \frac{3}{8} x^{2}$

Step 1.5

Multiply $- \frac{1}{2} (x y)$ .

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Step 1.5.1

Combine $x$ and $\frac{1}{2}$ .

$\frac{x^{2}}{4} - \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{x}{2} y + \frac{1}{4} y^{2} - \frac{3}{8} x^{2}$

Step 1.5.2

Combine $y$ and $\frac{x}{2}$ .

$\frac{x^{2}}{4} - \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{y x}{2} + \frac{1}{4} y^{2} - \frac{3}{8} x^{2}$

Step 1.6

Combine $\frac{1}{4}$ and $y^{2}$ .

$\frac{x^{2}}{4} - \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{y x}{2} + \frac{y^{2}}{4} - \frac{3}{8} x^{2}$

Step 1.7

Combine $x^{2}$ and $\frac{3}{8}$ .

$\frac{x^{2}}{4} - \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{y x}{2} + \frac{y^{2}}{4} - \frac{x^{2} \cdot 3}{8}$

Step 1.8

Move $3$ to the left of $x^{2}$ .

$\frac{x^{2}}{4} - \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{y x}{2} + \frac{y^{2}}{4} - \frac{3 x^{2}}{8}$

Step 2

To write $\frac{x^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{y x}{2} + \frac{y^{2}}{4} + \frac{x^{2}}{4} \cdot \frac{2}{2} - \frac{3 x^{2}}{8}$

Step 3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x^{2}}{4}$ by $\frac{2}{2}$ .

$- \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{y x}{2} + \frac{y^{2}}{4} + \frac{x^{2} \cdot 2}{4 \cdot 2} - \frac{3 x^{2}}{8}$

Step 3.2

Multiply $4$ by $2$ .

$- \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{y x}{2} + \frac{y^{2}}{4} + \frac{x^{2} \cdot 2}{8} - \frac{3 x^{2}}{8}$

Step 4

Simplify terms.

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Step 4.1

Combine the numerators over the common denominator.

$- \frac{3 y x}{8} + \frac{y^{2}}{2} - \frac{y x}{2} + \frac{y^{2}}{4} + \frac{x^{2} \cdot 2 - 3 x^{2}}{8}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{- 3 y x + x^{2} \cdot 2 - 3 x^{2}}{8} + \frac{y^{2}}{2} + \frac{- y x}{2} + \frac{y^{2}}{4}$

Step 5

Move $2$ to the left of $x^{2}$ .

$\frac{- 3 y x + 2 x^{2} - 3 x^{2}}{8} + \frac{y^{2}}{2} + \frac{- y x}{2} + \frac{y^{2}}{4}$

Step 6

Subtract $3 x^{2}$ from $2 x^{2}$ .

$\frac{- 3 y x - x^{2}}{8} + \frac{y^{2}}{2} + \frac{- y x}{2} + \frac{y^{2}}{4}$

Step 7

Simplify each term.

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Step 7.1

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

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Step 7.1.1

Factor $x$ out of $- 3 y x$ .

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

Step 7.1.2

Factor $x$ out of $- x^{2}$ .

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

Step 7.1.3

Factor $x$ out of $x (- 3 y) + x (- x)$ .

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

Step 7.2

Move the negative in front of the fraction.

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

Step 8

To write $\frac{y^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 9

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.1

Multiply $\frac{y^{2}}{2}$ by $\frac{4}{4}$ .

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

Step 9.2

Multiply $2$ by $4$ .

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify the numerator.

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Step 11.1

Apply the distributive property.

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

Step 11.2

Rewrite using the commutative property of multiplication.

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

Step 11.3

Rewrite using the commutative property of multiplication.

$\frac{- 3 x y - x \cdot x + y^{2} \cdot 4}{8} - \frac{y x}{2} + \frac{y^{2}}{4}$

Step 11.4

Multiply $x$ by $x$ by adding the exponents.

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Step 11.4.1

Move $x$ .

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

Step 11.4.2

Multiply $x$ by $x$ .

$\frac{- 3 x y - x^{2} + y^{2} \cdot 4}{8} - \frac{y x}{2} + \frac{y^{2}}{4}$

Step 11.5

Move $4$ to the left of $y^{2}$ .

$\frac{- 3 x y - x^{2} + 4 \cdot y^{2}}{8} - \frac{y x}{2} + \frac{y^{2}}{4}$

Step 11.6

Factor by grouping.

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Step 11.6.1

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 $b = - 3$ .

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Step 11.6.1.1

Reorder terms.

$\frac{- x^{2} + 4 y^{2} - 3 x y}{8} - \frac{y x}{2} + \frac{y^{2}}{4}$

Step 11.6.1.2

Reorder $4 y^{2}$ and $- 3 x y$ .

$\frac{- x^{2} - 3 x y + 4 y^{2}}{8} - \frac{y x}{2} + \frac{y^{2}}{4}$

Step 11.6.1.3

Factor $- 3$ out of $- 3 x y$ .

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

Step 11.6.1.4

Rewrite $- 3$ as $1$ plus $- 4$

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

Step 11.6.1.5

Apply the distributive property.

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

Step 11.6.1.6

Multiply $x y$ by $1$ .

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

Step 11.6.1.7

Move parentheses.

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

Step 11.6.2

Factor out the greatest common factor from each group.

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Step 11.6.2.1

Group the first two terms and the last two terms.

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

Step 11.6.2.2

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

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

Step 11.6.3

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

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

Step 12

To write $- \frac{y x}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{(- x + y) (x + 4 y)}{8} - \frac{y x}{2} \cdot \frac{4}{4} + \frac{y^{2}}{4}$

Step 13

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 13.1

Multiply $\frac{y x}{2}$ by $\frac{4}{4}$ .

$\frac{(- x + y) (x + 4 y)}{8} - \frac{y x \cdot 4}{2 \cdot 4} + \frac{y^{2}}{4}$

Step 13.2

Multiply $2$ by $4$ .

$\frac{(- x + y) (x + 4 y)}{8} - \frac{y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 14

Combine the numerators over the common denominator.

$\frac{(- x + y) (x + 4 y) - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15

Simplify the numerator.

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Step 15.1

Expand $(- x + y) (x + 4 y)$ using the FOIL Method.

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Step 15.1.1

Apply the distributive property.

$\frac{- x (x + 4 y) + y (x + 4 y) - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.1.2

Apply the distributive property.

$\frac{- x \cdot x - x (4 y) + y (x + 4 y) - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.1.3

Apply the distributive property.

$\frac{- x \cdot x - x (4 y) + y x + y (4 y) - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.2

Simplify and combine like terms.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 15.2.1.1.1

Move $x$ .

$\frac{- (x \cdot x) - x (4 y) + y x + y (4 y) - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.2.1.1.2

Multiply $x$ by $x$ .

$\frac{- x^{2} - x (4 y) + y x + y (4 y) - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.2.1.2

Rewrite using the commutative property of multiplication.

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

Step 15.2.1.3

Multiply $- 1$ by $4$ .

$\frac{- x^{2} - 4 x y + y x + y (4 y) - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{- x^{2} - 4 x y + y x + 4 y \cdot y - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.2.1.5

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

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Step 15.2.1.5.1

Move $y$ .

$\frac{- x^{2} - 4 x y + y x + 4 (y \cdot y) - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.2.1.5.2

Multiply $y$ by $y$ .

$\frac{- x^{2} - 4 x y + y x + 4 y^{2} - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.2.2

Add $- 4 x y$ and $y x$ .

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Step 15.2.2.1

Reorder $y$ and $x$ .

$\frac{- x^{2} - 4 x y + x y + 4 y^{2} - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.2.2.2

Add $- 4 x y$ and $x y$ .

$\frac{- x^{2} - 3 x y + 4 y^{2} - y x \cdot 4}{8} + \frac{y^{2}}{4}$

Step 15.3

Multiply $4$ by $- 1$ .

$\frac{- x^{2} - 3 x y + 4 y^{2} - 4 y x}{8} + \frac{y^{2}}{4}$

Step 15.4

Subtract $4 y x$ from $- 3 x y$ .

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Step 15.4.1

Move $y$ .

$\frac{- x^{2} + 4 y^{2} - 3 x y - 4 x y}{8} + \frac{y^{2}}{4}$

Step 15.4.2

Subtract $4 x y$ from $- 3 x y$ .

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

Step 16

To write $\frac{y^{2}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 17

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 17.1

Multiply $\frac{y^{2}}{4}$ by $\frac{2}{2}$ .

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

Step 17.2

Multiply $4$ by $2$ .

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

Step 18

Combine the numerators over the common denominator.

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

Step 19

Simplify the numerator.

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Step 19.1

Move $2$ to the left of $y^{2}$ .

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

Step 19.2

Add $4 y^{2}$ and $2 y^{2}$ .

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

Step 20

Simplify with factoring out.

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Step 20.1

Factor $- 1$ out of $- x^{2}$ .

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

Step 20.2

Factor $- 1$ out of $6 y^{2}$ .

$\frac{- (x^{2}) - (- 6 y^{2}) - 7 x y}{8}$

Step 20.3

Factor $- 1$ out of $- (x^{2}) - (- 6 y^{2})$ .

$\frac{- (x^{2} - 6 y^{2}) - 7 x y}{8}$

Step 20.4

Factor $- 1$ out of $- 7 x y$ .

$\frac{- (x^{2} - 6 y^{2}) - (7 x y)}{8}$

Step 20.5

Factor $- 1$ out of $- (x^{2} - 6 y^{2}) - (7 x y)$ .

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

Step 20.6

Simplify the expression.

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Step 20.6.1

Rewrite $- (x^{2} - 6 y^{2} + 7 x y)$ as $- 1 (x^{2} - 6 y^{2} + 7 x y)$ .

$\frac{- 1 (x^{2} - 6 y^{2} + 7 x y)}{8}$

Step 20.6.2

Move the negative in front of the fraction.

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