Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

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

Step 1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.5

Multiply $y$ by $y$ .

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

Step 1.3.2

Add $2 x y$ and $2 y x$ .

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Step 1.3.2.1

Move $y$ .

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

Step 1.3.2.2

Add $2 x y$ and $2 x y$ .

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

Step 1.4

Rewrite ${(3 x - 2 y)}^{2}$ as $(3 x - 2 y) (3 x - 2 y)$ .

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

Step 1.5

Expand $(3 x - 2 y) (3 x - 2 y)$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.6.1.2

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

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Step 1.6.1.2.1

Move $x$ .

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

Step 1.6.1.2.2

Multiply $x$ by $x$ .

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

Step 1.6.1.3

Multiply $3$ by $3$ .

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} + 3 x (- 2 y) - 2 y (3 x) - 2 y (- 2 y)) + (x - 4 y) (x + 4 y)$

Step 1.6.1.4

Rewrite using the commutative property of multiplication.

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} + 3 \cdot - 2 x y - 2 y (3 x) - 2 y (- 2 y)) + (x - 4 y) (x + 4 y)$

Step 1.6.1.5

Multiply $3$ by $- 2$ .

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 6 x y - 2 y (3 x) - 2 y (- 2 y)) + (x - 4 y) (x + 4 y)$

Step 1.6.1.6

Rewrite using the commutative property of multiplication.

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 6 x y - 2 \cdot 3 y x - 2 y (- 2 y)) + (x - 4 y) (x + 4 y)$

Step 1.6.1.7

Multiply $- 2$ by $3$ .

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 6 x y - 6 y x - 2 y (- 2 y)) + (x - 4 y) (x + 4 y)$

Step 1.6.1.8

Rewrite using the commutative property of multiplication.

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 6 x y - 6 y x - 2 \cdot - 2 y \cdot y) + (x - 4 y) (x + 4 y)$

Step 1.6.1.9

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

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Step 1.6.1.9.1

Move $y$ .

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 6 x y - 6 y x - 2 \cdot - 2 (y \cdot y)) + (x - 4 y) (x + 4 y)$

Step 1.6.1.9.2

Multiply $y$ by $y$ .

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 6 x y - 6 y x - 2 \cdot - 2 y^{2}) + (x - 4 y) (x + 4 y)$

Step 1.6.1.10

Multiply $- 2$ by $- 2$ .

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 6 x y - 6 y x + 4 y^{2}) + (x - 4 y) (x + 4 y)$

Step 1.6.2

Subtract $6 y x$ from $- 6 x y$ .

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Step 1.6.2.1

Move $y$ .

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 6 x y - 6 x y + 4 y^{2}) + (x - 4 y) (x + 4 y)$

Step 1.6.2.2

Subtract $6 x y$ from $- 6 x y$ .

$4 x^{2} + 4 x y + y^{2} - (9 x^{2} - 12 x y + 4 y^{2}) + (x - 4 y) (x + 4 y)$

Step 1.7

Apply the distributive property.

$4 x^{2} + 4 x y + y^{2} - (9 x^{2}) - (- 12 x y) - (4 y^{2}) + (x - 4 y) (x + 4 y)$

Step 1.8

Simplify.

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Step 1.8.1

Multiply $9$ by $- 1$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} - (- 12 x y) - (4 y^{2}) + (x - 4 y) (x + 4 y)$

Step 1.8.2

Multiply $- 12$ by $- 1$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 (x y) - (4 y^{2}) + (x - 4 y) (x + 4 y)$

Step 1.8.3

Multiply $4$ by $- 1$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 (x y) - 4 y^{2} + (x - 4 y) (x + 4 y)$

Step 1.9

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

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Step 1.9.1

Apply the distributive property.

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x (x + 4 y) - 4 y (x + 4 y)$

Step 1.9.2

Apply the distributive property.

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x \cdot x + x (4 y) - 4 y (x + 4 y)$

Step 1.9.3

Apply the distributive property.

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x \cdot x + x (4 y) - 4 y x - 4 y (4 y)$

Step 1.10

Combine the opposite terms in $x \cdot x + x (4 y) - 4 y x - 4 y (4 y)$ .

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Step 1.10.1

Reorder the factors in the terms $x (4 y)$ and $- 4 y x$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x \cdot x + 4 x y - 4 x y - 4 y (4 y)$

Step 1.10.2

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

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x \cdot x + 0 - 4 y (4 y)$

Step 1.10.3

Add $x \cdot x$ and $0$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x \cdot x - 4 y (4 y)$

Step 1.11

Simplify each term.

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Step 1.11.1

Multiply $x$ by $x$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x^{2} - 4 y (4 y)$

Step 1.11.2

Rewrite using the commutative property of multiplication.

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x^{2} - 4 \cdot 4 y \cdot y$

Step 1.11.3

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

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Step 1.11.3.1

Move $y$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x^{2} - 4 \cdot 4 (y \cdot y)$

Step 1.11.3.2

Multiply $y$ by $y$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x^{2} - 4 \cdot 4 y^{2}$

Step 1.11.4

Multiply $- 4$ by $4$ .

$4 x^{2} + 4 x y + y^{2} - 9 x^{2} + 12 x y - 4 y^{2} + x^{2} - 16 y^{2}$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $9 x^{2}$ from $4 x^{2}$ .

$- 5 x^{2} + 4 x y + y^{2} + 12 x y - 4 y^{2} + x^{2} - 16 y^{2}$

Step 2.2

Add $- 5 x^{2}$ and $x^{2}$ .

$- 4 x^{2} + 4 x y + y^{2} + 12 x y - 4 y^{2} - 16 y^{2}$

Step 2.3

Add $4 x y$ and $12 x y$ .

$- 4 x^{2} + y^{2} + 16 x y - 4 y^{2} - 16 y^{2}$

Step 2.4

Subtract $4 y^{2}$ from $y^{2}$ .

$- 4 x^{2} - 3 y^{2} + 16 x y - 16 y^{2}$

Step 2.5

Subtract $16 y^{2}$ from $- 3 y^{2}$ .

$- 4 x^{2} - 19 y^{2} + 16 x y$