Basic Math Examples

${(2 a - \frac{1}{3})}^{2} - {(a - \frac{5}{3})}^{2}$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a - \frac{1}{3}$ and $b = a - \frac{5}{3}$ .

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

Step 2

Simplify.

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Step 2.1

Add $2 a$ and $a$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Subtract $5$ from $- 1$ .

$(3 a + \frac{- 6}{3}) (2 a - \frac{1}{3} - (a - \frac{5}{3}))$

Step 2.4

Factor $3$ out of $3 a + \frac{- 6}{3}$ .

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Step 2.4.1

Factor $3$ out of $3 a$ .

$(3 (a) + \frac{- 6}{3}) (2 a - \frac{1}{3} - (a - \frac{5}{3}))$

Step 2.4.2

Factor $3$ out of $\frac{- 6}{3}$ .

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

Step 2.4.3

Factor $3$ out of $3 a + 3 (\frac{- 2}{3})$ .

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

Step 2.5

Apply the distributive property.

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

Step 2.6

Multiply $- - \frac{5}{3}$ .

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Step 2.6.1

Multiply $- 1$ by $- 1$ .

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

Step 2.6.2

Multiply $\frac{5}{3}$ by $1$ .

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

Step 2.7

Subtract $a$ from $2 a$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Add $- 1$ and $5$ .

$3 (a + \frac{- 2}{3}) (a + \frac{4}{3})$