Algebra Examples

$x {(2 y - x)}^{2} + 2 x^{2} (x - 2 y)$

Step 1

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

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Step 1.1

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

$x ({(2 y - x)}^{2}) + 2 x^{2} (x - 2 y)$

Step 1.2

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

$x ({(2 y - x)}^{2}) + x (2 x (x - 2 y))$

Step 1.3

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

$x ({(2 y - x)}^{2} + 2 x (x - 2 y))$

Step 2

Rewrite ${(2 y - x)}^{2}$ as $(2 y - x) (2 y - x)$ .

$x ((2 y - x) (2 y - x) + 2 x (x - 2 y))$

Step 3

Expand $(2 y - x) (2 y - x)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$x (2 y (2 y - x) - x (2 y - x) + 2 x (x - 2 y))$

Step 3.2

Apply the distributive property.

$x (2 y (2 y) + 2 y (- x) - x (2 y - x) + 2 x (x - 2 y))$

Step 3.3

Apply the distributive property.

$x (2 y (2 y) + 2 y (- x) - x (2 y) - x (- x) + 2 x (x - 2 y))$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

$x (2 \cdot 2 y \cdot y + 2 y (- x) - x (2 y) - x (- x) + 2 x (x - 2 y))$

Step 4.1.2

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

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Step 4.1.2.1

Move $y$ .

$x (2 \cdot 2 (y \cdot y) + 2 y (- x) - x (2 y) - x (- x) + 2 x (x - 2 y))$

Step 4.1.2.2

Multiply $y$ by $y$ .

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

Step 4.1.3

Multiply $2$ by $2$ .

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

Step 4.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.1.5

Multiply $2$ by $- 1$ .

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

Step 4.1.6

Rewrite using the commutative property of multiplication.

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

Step 4.1.7

Multiply $- 1$ by $2$ .

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

Step 4.1.8

Rewrite using the commutative property of multiplication.

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

Step 4.1.9

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

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Step 4.1.9.1

Move $x$ .

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

Step 4.1.9.2

Multiply $x$ by $x$ .

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

Step 4.1.10

Multiply $- 1$ by $- 1$ .

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

Step 4.1.11

Multiply $x^{2}$ by $1$ .

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

Step 4.2

Subtract $2 x y$ from $- 2 y x$ .

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Step 4.2.1

Move $y$ .

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

Step 4.2.2

Subtract $2 x y$ from $- 2 x y$ .

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

Move $x$ .

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 7

Rewrite using the commutative property of multiplication.

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

Step 8

Multiply $2$ by $- 2$ .

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

Step 9

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

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

Step 10

Add $x^{2}$ and $2 x^{2}$ .

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

Step 11

Factor.

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Step 11.1

Factor by grouping.

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Step 11.1.1

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 $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = - 8$ .

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Step 11.1.1.1

Reorder terms.

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

Step 11.1.1.2

Reorder $4 y^{2}$ and $- 8 x y$ .

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

Step 11.1.1.3

Factor $- 8$ out of $- 8 x y$ .

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

Step 11.1.1.4

Rewrite $- 8$ as $- 2$ plus $- 6$

$x (3 x^{2} + (- 2 - 6) (x y) + 4 y^{2})$

Step 11.1.1.5

Apply the distributive property.

$x (3 x^{2} - 2 (x y) - 6 (x y) + 4 y^{2})$

Step 11.1.1.6

Remove unnecessary parentheses.

$x (3 x^{2} - 2 x y - 6 (x y) + 4 y^{2})$

Step 11.1.1.7

Remove unnecessary parentheses.

$x (3 x^{2} - 2 x y - 6 x y + 4 y^{2})$

Step 11.1.2

Factor out the greatest common factor from each group.

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Step 11.1.2.1

Group the first two terms and the last two terms.

$x ((3 x^{2} - 2 x y) - 6 x y + 4 y^{2})$

Step 11.1.2.2

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

$x (x (3 x - 2 y) - 2 y (3 x - 2 y))$

Step 11.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2 y$ .

$x ((3 x - 2 y) (x - 2 y))$

Step 11.2

Remove unnecessary parentheses.

$x (3 x - 2 y) (x - 2 y)$