Algebra Examples

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

Step 1

Rewrite the middle term.

$9 {(x - y)}^{2} + 2 (3 (x - y)) (2 (x + y)) + 0 + 4 {(x + y)}^{2}$

Step 2

Rearrange terms.

$9 {(x - y)}^{2} + 2 (3 (x - y)) (2 (x + y)) + 4 {(x + y)}^{2} + 0$

Step 3

Factor first three terms by perfect square rule.

${(3 (x - y) + 2 (x + y))}^{2} + 0$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

${(3 x + 3 (- y) + 2 (x + y))}^{2} + 0$

Step 4.2

Multiply $- 1$ by $3$ .

${(3 x - 3 y + 2 (x + y))}^{2} + 0$

Step 4.3

Apply the distributive property.

${(3 x - 3 y + 2 x + 2 y)}^{2} + 0$

Step 5

Add $3 x$ and $2 x$ .

${(5 x - 3 y + 2 y)}^{2} + 0$

Step 6

Add $- 3 y$ and $2 y$ .

${(5 x - y)}^{2} + 0$

Step 7

Rewrite ${(5 x - y)}^{2}$ as $(5 x - y) (5 x - y)$ .

$(5 x - y) (5 x - y) + 0$

Step 8

Expand $(5 x - y) (5 x - y)$ using the FOIL Method.

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Step 8.1

Apply the distributive property.

$5 x (5 x - y) - y (5 x - y) + 0$

Step 8.2

Apply the distributive property.

$5 x (5 x) + 5 x (- y) - y (5 x - y) + 0$

Step 8.3

Apply the distributive property.

$5 x (5 x) + 5 x (- y) - y (5 x) - y (- y) + 0$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Rewrite using the commutative property of multiplication.

$5 \cdot 5 x \cdot x + 5 x (- y) - y (5 x) - y (- y) + 0$

Step 9.1.2

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

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Step 9.1.2.1

Move $x$ .

$5 \cdot 5 (x \cdot x) + 5 x (- y) - y (5 x) - y (- y) + 0$

Step 9.1.2.2

Multiply $x$ by $x$ .

$5 \cdot 5 x^{2} + 5 x (- y) - y (5 x) - y (- y) + 0$

Step 9.1.3

Multiply $5$ by $5$ .

$25 x^{2} + 5 x (- y) - y (5 x) - y (- y) + 0$

Step 9.1.4

Rewrite using the commutative property of multiplication.

$25 x^{2} + 5 \cdot - 1 x y - y (5 x) - y (- y) + 0$

Step 9.1.5

Multiply $5$ by $- 1$ .

$25 x^{2} - 5 x y - y (5 x) - y (- y) + 0$

Step 9.1.6

Rewrite using the commutative property of multiplication.

$25 x^{2} - 5 x y - 1 \cdot 5 y x - y (- y) + 0$

Step 9.1.7

Multiply $- 1$ by $5$ .

$25 x^{2} - 5 x y - 5 y x - y (- y) + 0$

Step 9.1.8

Rewrite using the commutative property of multiplication.

$25 x^{2} - 5 x y - 5 y x - 1 \cdot - 1 y \cdot y + 0$

Step 9.1.9

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

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Step 9.1.9.1

Move $y$ .

$25 x^{2} - 5 x y - 5 y x - 1 \cdot - 1 (y \cdot y) + 0$

Step 9.1.9.2

Multiply $y$ by $y$ .

$25 x^{2} - 5 x y - 5 y x - 1 \cdot - 1 y^{2} + 0$

Step 9.1.10

Multiply $- 1$ by $- 1$ .

$25 x^{2} - 5 x y - 5 y x + 1 y^{2} + 0$

Step 9.1.11

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

$25 x^{2} - 5 x y - 5 y x + y^{2} + 0$

Step 9.2

Subtract $5 y x$ from $- 5 x y$ .

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Step 9.2.1

Move $y$ .

$25 x^{2} - 5 x y - 5 x y + y^{2} + 0$

Step 9.2.2

Subtract $5 x y$ from $- 5 x y$ .

$25 x^{2} - 10 x y + y^{2} + 0$

Step 10

Add $25 x^{2} - 10 x y + y^{2}$ and $0$ .

$25 x^{2} - 10 x y + y^{2}$

Step 11

Factor using the perfect square rule.

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Step 11.1

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

${(5 x)}^{2} - 10 x y + y^{2}$

Step 11.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x y = 2 \cdot (5 x) \cdot y$

Step 11.3

Rewrite the polynomial.

${(5 x)}^{2} - 2 \cdot (5 x) \cdot y + y^{2}$

Step 11.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 5 x$ and $b = y$ .

${(5 x - y)}^{2}$