Algebra Examples

$2 - \frac{x^{2} + 10 x y + 7 y^{2}}{x^{2} + 4 x y + 4 y^{2}}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$2 - \frac{x^{2} + 10 x y + 7 y^{2}}{x^{2} + 4 x y + {(2 y)}^{2}}$

Step 1.2

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

$4 x y = 2 \cdot x \cdot (2 y)$

Step 1.3

Rewrite the polynomial.

$2 - \frac{x^{2} + 10 x y + 7 y^{2}}{x^{2} + 2 \cdot x \cdot (2 y) + {(2 y)}^{2}}$

Step 1.4

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

$2 - \frac{x^{2} + 10 x y + 7 y^{2}}{{(x + 2 y)}^{2}}$

Step 2

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

$\frac{2 {(x + 2 y)}^{2}}{{(x + 2 y)}^{2}} - \frac{x^{2} + 10 x y + 7 y^{2}}{{(x + 2 y)}^{2}}$

Step 3

Combine the numerators over the common denominator.

$\frac{2 {(x + 2 y)}^{2} - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4

Rewrite $\frac{2 {(x + 2 y)}^{2} - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$ in a factored form.

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Step 4.1

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

$\frac{2 ((x + 2 y) (x + 2 y)) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

$\frac{2 (x (x + 2 y) + 2 y (x + 2 y)) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.2.2

Apply the distributive property.

$\frac{2 (x \cdot x + x (2 y) + 2 y (x + 2 y)) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.2.3

Apply the distributive property.

$\frac{2 (x \cdot x + x (2 y) + 2 y x + 2 y (2 y)) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $x$ by $x$ .

$\frac{2 (x^{2} + x (2 y) + 2 y x + 2 y (2 y)) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.3.1.2

Rewrite using the commutative property of multiplication.

$\frac{2 (x^{2} + 2 x y + 2 y x + 2 y (2 y)) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{2 (x^{2} + 2 x y + 2 y x + 2 \cdot 2 y \cdot y) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.3.1.4

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

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Step 4.3.1.4.1

Move $y$ .

$\frac{2 (x^{2} + 2 x y + 2 y x + 2 \cdot 2 (y \cdot y)) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.3.1.4.2

Multiply $y$ by $y$ .

$\frac{2 (x^{2} + 2 x y + 2 y x + 2 \cdot 2 y^{2}) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.3.1.5

Multiply $2$ by $2$ .

$\frac{2 (x^{2} + 2 x y + 2 y x + 4 y^{2}) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.3.2

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

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Step 4.3.2.1

Move $y$ .

$\frac{2 (x^{2} + 2 x y + 2 x y + 4 y^{2}) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.3.2.2

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

$\frac{2 (x^{2} + 4 x y + 4 y^{2}) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.4

Apply the distributive property.

$\frac{2 x^{2} + 2 (4 x y) + 2 (4 y^{2}) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.5

Simplify.

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Step 4.5.1

Multiply $4$ by $2$ .

$\frac{2 x^{2} + 8 (x y) + 2 (4 y^{2}) - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.5.2

Multiply $4$ by $2$ .

$\frac{2 x^{2} + 8 (x y) + 8 y^{2} - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.6

Remove parentheses.

$\frac{2 x^{2} + 8 x y + 8 y^{2} - (x^{2} + 10 x y + 7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.7

Apply the distributive property.

$\frac{2 x^{2} + 8 x y + 8 y^{2} - x^{2} - (10 x y) - (7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.8

Simplify.

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Step 4.8.1

Multiply $10$ by $- 1$ .

$\frac{2 x^{2} + 8 x y + 8 y^{2} - x^{2} - 10 (x y) - (7 y^{2})}{{(x + 2 y)}^{2}}$

Step 4.8.2

Multiply $7$ by $- 1$ .

$\frac{2 x^{2} + 8 x y + 8 y^{2} - x^{2} - 10 (x y) - 7 y^{2}}{{(x + 2 y)}^{2}}$

Step 4.9

Remove parentheses.

$\frac{2 x^{2} + 8 x y + 8 y^{2} - x^{2} - 10 x y - 7 y^{2}}{{(x + 2 y)}^{2}}$

Step 4.10

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

$\frac{x^{2} + 8 x y + 8 y^{2} - 10 x y - 7 y^{2}}{{(x + 2 y)}^{2}}$

Step 4.11

Subtract $10 x y$ from $8 x y$ .

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

Step 4.12

Subtract $7 y^{2}$ from $8 y^{2}$ .

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

Step 4.13

Factor using the perfect square rule.

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Step 4.13.1

Rearrange terms.

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

Step 4.13.2

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

$2 x y = 2 \cdot x \cdot y$

Step 4.13.3

Rewrite the polynomial.

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

Step 4.13.4

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

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