Basic Math Examples

$\frac{25 - a^{2}}{b + 3} \cdot \frac{2 b^{2} + 6 b}{a - 5}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{5^{2} - a^{2}}{b + 3} \cdot \frac{2 b^{2} + 6 b}{a - 5}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = a$ .

$\frac{(5 + a) (5 - a)}{b + 3} \cdot \frac{2 b^{2} + 6 b}{a - 5}$

Step 2

Simplify terms.

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Step 2.1

Factor $2 b$ out of $2 b^{2} + 6 b$ .

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Step 2.1.1

Factor $2 b$ out of $2 b^{2}$ .

$\frac{(5 + a) (5 - a)}{b + 3} \cdot \frac{2 b (b) + 6 b}{a - 5}$

Step 2.1.2

Factor $2 b$ out of $6 b$ .

$\frac{(5 + a) (5 - a)}{b + 3} \cdot \frac{2 b (b) + 2 b (3)}{a - 5}$

Step 2.1.3

Factor $2 b$ out of $2 b (b) + 2 b (3)$ .

$\frac{(5 + a) (5 - a)}{b + 3} \cdot \frac{2 b (b + 3)}{a - 5}$

Step 2.2

Cancel the common factor of $b + 3$ .

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Step 2.2.1

Factor $b + 3$ out of $2 b (b + 3)$ .

$\frac{(5 + a) (5 - a)}{b + 3} \cdot \frac{(b + 3) (2 b)}{a - 5}$

Step 2.2.2

Cancel the common factor.

$\frac{(5 + a) (5 - a)}{b + 3} \cdot \frac{(b + 3) (2 b)}{a - 5}$

Step 2.2.3

Rewrite the expression.

$(5 + a) (5 - a) \cdot \frac{2 b}{a - 5}$

Step 3

Expand $(5 + a) (5 - a)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$(5 (5 - a) + a (5 - a)) \cdot \frac{2 b}{a - 5}$

Step 3.2

Apply the distributive property.

$(5 \cdot 5 + 5 (- a) + a (5 - a)) \cdot \frac{2 b}{a - 5}$

Step 3.3

Apply the distributive property.

$(5 \cdot 5 + 5 (- a) + a \cdot 5 + a (- a)) \cdot \frac{2 b}{a - 5}$

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

Multiply $5$ by $5$ .

$(25 + 5 (- a) + a \cdot 5 + a (- a)) \cdot \frac{2 b}{a - 5}$

Step 4.1.2

Multiply $- 1$ by $5$ .

$(25 - 5 a + a \cdot 5 + a (- a)) \cdot \frac{2 b}{a - 5}$

Step 4.1.3

Move $5$ to the left of $a$ .

$(25 - 5 a + 5 \cdot a + a (- a)) \cdot \frac{2 b}{a - 5}$

Step 4.1.4

Rewrite using the commutative property of multiplication.

$(25 - 5 a + 5 a - a \cdot a) \cdot \frac{2 b}{a - 5}$

Step 4.1.5

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1.5.1

Move $a$ .

$(25 - 5 a + 5 a - (a \cdot a)) \cdot \frac{2 b}{a - 5}$

Step 4.1.5.2

Multiply $a$ by $a$ .

$(25 - 5 a + 5 a - a^{2}) \cdot \frac{2 b}{a - 5}$

Step 4.2

Add $- 5 a$ and $5 a$ .

$(25 + 0 - a^{2}) \cdot \frac{2 b}{a - 5}$

Step 4.3

Add $25$ and $0$ .

$(25 - a^{2}) \cdot \frac{2 b}{a - 5}$

Step 5

Multiply $25 - a^{2}$ by $\frac{2 b}{a - 5}$ .

$\frac{(25 - a^{2}) (2 b)}{a - 5}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $25$ as $5^{2}$ .

$\frac{(5^{2} - a^{2}) \cdot 2 b}{a - 5}$

Step 6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = a$ .

$\frac{(5 + a) (5 - a) \cdot 2 b}{a - 5}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $5 - a$ and $a - 5$ .

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Step 7.1.1

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{(5 + a) (- 1 (- 5) - a) \cdot 2 b}{a - 5}$

Step 7.1.2

Factor $- 1$ out of $- a$ .

$\frac{(5 + a) (- 1 (- 5) - (a)) \cdot 2 b}{a - 5}$

Step 7.1.3

Factor $- 1$ out of $- 1 (- 5) - (a)$ .

$\frac{(5 + a) (- 1 (- 5 + a)) \cdot 2 b}{a - 5}$

Step 7.1.4

Reorder terms.

$\frac{(5 + a) (- 1 (a - 5)) \cdot 2 b}{a - 5}$

Step 7.1.5

Cancel the common factor.

$\frac{(5 + a) (- 1 (a - 5)) \cdot 2 b}{a - 5}$

Step 7.1.6

Divide $((5 + a) \cdot (- 1)) \cdot 2 b$ by $1$ .

$((5 + a) \cdot (- 1)) \cdot 2 b$

Step 7.2

Simplify by multiplying through.

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Step 7.2.1

Apply the distributive property.

$(5 \cdot - 1 + a \cdot - 1) \cdot 2 b$

Step 7.2.2

Simplify the expression.

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Step 7.2.2.1

Multiply $5$ by $- 1$ .

$(- 5 + a \cdot - 1) \cdot 2 b$

Step 7.2.2.2

Move $- 1$ to the left of $a$ .

$(- 5 - 1 \cdot a) \cdot 2 b$

Step 7.3

Rewrite $- 1 a$ as $- a$ .

$(- 5 - a) \cdot 2 b$

Step 7.4

Simplify by multiplying through.

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Step 7.4.1

Apply the distributive property.

$(- 5 \cdot 2 - a \cdot 2) b$

Step 7.4.2

Multiply.

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Step 7.4.2.1

Multiply $- 5$ by $2$ .

$(- 10 - a \cdot 2) b$

Step 7.4.2.2

Multiply $2$ by $- 1$ .

$(- 10 - 2 a) b$

Step 7.4.3

Apply the distributive property.

$- 10 b - 2 a b$