Finite Math Examples

$9 {(x - \frac{1}{2})}^{2} - 4 {(x + \frac{1}{2})}^{2}$

Step 1

Rewrite $9 {(x - \frac{1}{2})}^{2}$ as ${(3 (x - \frac{1}{2}))}^{2}$ .

${(3 (x - \frac{1}{2}))}^{2} - 4 {(x + \frac{1}{2})}^{2}$

Step 2

Rewrite $4 {(x + \frac{1}{2})}^{2}$ as ${(2 (x + \frac{1}{2}))}^{2}$ .

${(3 (x - \frac{1}{2}))}^{2} - {(2 (x + \frac{1}{2}))}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 (x - \frac{1}{2})$ and $b = 2 (x + \frac{1}{2})$ .

$(3 (x - \frac{1}{2}) + 2 (x + \frac{1}{2})) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(3 x + 3 (- \frac{1}{2}) + 2 (x + \frac{1}{2})) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.2

Multiply $3 (- \frac{1}{2})$ .

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Step 4.2.1

Multiply $- 1$ by $3$ .

$(3 x - 3 (\frac{1}{2}) + 2 (x + \frac{1}{2})) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.2.2

Combine $- 3$ and $\frac{1}{2}$ .

$(3 x + \frac{- 3}{2} + 2 (x + \frac{1}{2})) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.3

Move the negative in front of the fraction.

$(3 x - \frac{3}{2} + 2 (x + \frac{1}{2})) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.4

Apply the distributive property.

$(3 x - \frac{3}{2} + 2 x + 2 (\frac{1}{2})) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Cancel the common factor.

$(3 x - \frac{3}{2} + 2 x + 2 (\frac{1}{2})) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.5.2

Rewrite the expression.

$(3 x - \frac{3}{2} + 2 x + 1) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.6

Add $3 x$ and $2 x$ .

$(5 x - \frac{3}{2} + 1) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.7

Write $1$ as a fraction with a common denominator.

$(5 x - \frac{3}{2} + \frac{2}{2}) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.8

Combine the numerators over the common denominator.

$(5 x + \frac{- 3 + 2}{2}) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.9

Add $- 3$ and $2$ .

$(5 x + \frac{- 1}{2}) (3 (x - \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.10

Apply the distributive property.

$(5 x + \frac{- 1}{2}) (3 x + 3 (- \frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.11

Multiply $3 (- \frac{1}{2})$ .

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Step 4.11.1

Multiply $- 1$ by $3$ .

$(5 x + \frac{- 1}{2}) (3 x - 3 (\frac{1}{2}) - (2 (x + \frac{1}{2})))$

Step 4.11.2

Combine $- 3$ and $\frac{1}{2}$ .

$(5 x + \frac{- 1}{2}) (3 x + \frac{- 3}{2} - (2 (x + \frac{1}{2})))$

Step 4.12

Move the negative in front of the fraction.

$(5 x + \frac{- 1}{2}) (3 x - \frac{3}{2} - (2 (x + \frac{1}{2})))$

Step 4.13

Apply the distributive property.

$(5 x + \frac{- 1}{2}) (3 x - \frac{3}{2} - (2 x + 2 (\frac{1}{2})))$

Step 4.14

Cancel the common factor of $2$ .

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Step 4.14.1

Cancel the common factor.

$(5 x + \frac{- 1}{2}) (3 x - \frac{3}{2} - (2 x + 2 (\frac{1}{2})))$

Step 4.14.2

Rewrite the expression.

$(5 x + \frac{- 1}{2}) (3 x - \frac{3}{2} - (2 x + 1))$

Step 4.15

Apply the distributive property.

$(5 x + \frac{- 1}{2}) (3 x - \frac{3}{2} - (2 x) - 1 \cdot 1)$

Step 4.16

Multiply $2$ by $- 1$ .

$(5 x + \frac{- 1}{2}) (3 x - \frac{3}{2} - 2 x - 1 \cdot 1)$

Step 4.17

Multiply $- 1$ by $1$ .

$(5 x + \frac{- 1}{2}) (3 x - \frac{3}{2} - 2 x - 1)$

Step 4.18

Subtract $2 x$ from $3 x$ .

$(5 x + \frac{- 1}{2}) (x - \frac{3}{2} - 1)$

Step 4.19

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(5 x + \frac{- 1}{2}) (x - \frac{3}{2} - 1 \cdot \frac{2}{2})$

Step 4.20

Combine $- 1$ and $\frac{2}{2}$ .

$(5 x + \frac{- 1}{2}) (x - \frac{3}{2} + \frac{- 1 \cdot 2}{2})$

Step 4.21

Combine the numerators over the common denominator.

$(5 x + \frac{- 1}{2}) (x + \frac{- 3 - 1 \cdot 2}{2})$

Step 4.22

Simplify the numerator.

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Step 4.22.1

Multiply $- 1$ by $2$ .

$(5 x + \frac{- 1}{2}) (x + \frac{- 3 - 2}{2})$

Step 4.22.2

Subtract $2$ from $- 3$ .

$(5 x + \frac{- 1}{2}) (x + \frac{- 5}{2})$