Basic Math Examples

$(3 a - \frac{1}{3} b) (a - \frac{1}{4} b)$

Step 1

Combine fractions.

Tap for more steps...

Step 1.1

Combine $b$ and $\frac{1}{3}$ .

$(3 a - \frac{b}{3}) (a - \frac{1}{4} b)$

Step 1.2

Combine $b$ and $\frac{1}{4}$ .

$(3 a - \frac{b}{3}) (a - \frac{b}{4})$

Step 2

Expand $(3 a - \frac{b}{3}) (a - \frac{b}{4})$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

$3 a (a - \frac{b}{4}) - \frac{b}{3} (a - \frac{b}{4})$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

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

Tap for more steps...

Step 3.1.1.1

Move $a$ .

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

Step 3.1.1.2

Multiply $a$ by $a$ .

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

Step 3.1.2

Multiply $3 a (- \frac{b}{4})$ .

Tap for more steps...

Step 3.1.2.1

Multiply $- 1$ by $3$ .

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

Step 3.1.2.2

Combine $- 3$ and $\frac{b}{4}$ .

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

Step 3.1.2.3

Combine $a$ and $\frac{- 3 b}{4}$ .

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

Step 3.1.3

Remove unnecessary parentheses.

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

Step 3.1.4

Move $- 3$ to the left of $a$ .

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

Step 3.1.5

Move the negative in front of the fraction.

$3 a^{2} - \frac{3 \cdot a b}{4} - \frac{b}{3} a - \frac{b}{3} (- \frac{b}{4})$

Step 3.1.6

Combine $a$ and $\frac{b}{3}$ .

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

Step 3.1.7

Multiply $- \frac{b}{3} (- \frac{b}{4})$ .

Tap for more steps...

Step 3.1.7.1

Multiply $- 1$ by $- 1$ .

$3 a^{2} - \frac{3 a b}{4} - \frac{a b}{3} + 1 \frac{b}{3} \frac{b}{4}$

Step 3.1.7.2

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

$3 a^{2} - \frac{3 a b}{4} - \frac{a b}{3} + \frac{b}{3} \cdot \frac{b}{4}$

Step 3.1.7.3

Multiply $\frac{b}{3}$ by $\frac{b}{4}$ .

$3 a^{2} - \frac{3 a b}{4} - \frac{a b}{3} + \frac{b \cdot b}{3 \cdot 4}$

Step 3.1.7.4

Raise $b$ to the power of $1$ .

$3 a^{2} - \frac{3 a b}{4} - \frac{a b}{3} + \frac{b^{1} b}{3 \cdot 4}$

Step 3.1.7.5

Raise $b$ to the power of $1$ .

$3 a^{2} - \frac{3 a b}{4} - \frac{a b}{3} + \frac{b^{1} b^{1}}{3 \cdot 4}$

Step 3.1.7.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 a^{2} - \frac{3 a b}{4} - \frac{a b}{3} + \frac{b^{1 + 1}}{3 \cdot 4}$

Step 3.1.7.7

Add $1$ and $1$ .

$3 a^{2} - \frac{3 a b}{4} - \frac{a b}{3} + \frac{b^{2}}{3 \cdot 4}$

Step 3.1.7.8

Multiply $3$ by $4$ .

$3 a^{2} - \frac{3 a b}{4} - \frac{a b}{3} + \frac{b^{2}}{12}$

Step 3.2

To write $- \frac{3 a b}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 a^{2} - \frac{3 a b}{4} \cdot \frac{3}{3} - \frac{a b}{3} + \frac{b^{2}}{12}$

Step 3.3

To write $- \frac{a b}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$3 a^{2} - \frac{3 a b}{4} \cdot \frac{3}{3} - \frac{a b}{3} \cdot \frac{4}{4} + \frac{b^{2}}{12}$

Step 3.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.4.1

Multiply $\frac{3 a b}{4}$ by $\frac{3}{3}$ .

$3 a^{2} - \frac{3 a b \cdot 3}{4 \cdot 3} - \frac{a b}{3} \cdot \frac{4}{4} + \frac{b^{2}}{12}$

Step 3.4.2

Multiply $4$ by $3$ .

$3 a^{2} - \frac{3 a b \cdot 3}{12} - \frac{a b}{3} \cdot \frac{4}{4} + \frac{b^{2}}{12}$

Step 3.4.3

Multiply $\frac{a b}{3}$ by $\frac{4}{4}$ .

$3 a^{2} - \frac{3 a b \cdot 3}{12} - \frac{a b \cdot 4}{3 \cdot 4} + \frac{b^{2}}{12}$

Step 3.4.4

Multiply $3$ by $4$ .

$3 a^{2} - \frac{3 a b \cdot 3}{12} - \frac{a b \cdot 4}{12} + \frac{b^{2}}{12}$

Step 3.5

Combine the numerators over the common denominator.

$3 a^{2} + \frac{- 3 a b \cdot 3 - a b \cdot 4}{12} + \frac{b^{2}}{12}$

Step 3.6

To write $3 a^{2}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$3 a^{2} \cdot \frac{12}{12} + \frac{- 3 a b \cdot 3 - a b \cdot 4}{12} + \frac{b^{2}}{12}$

Step 3.7

Combine $3 a^{2}$ and $\frac{12}{12}$ .

$\frac{3 a^{2} \cdot 12}{12} + \frac{- 3 a b \cdot 3 - a b \cdot 4}{12} + \frac{b^{2}}{12}$

Step 3.8

Combine the numerators over the common denominator.

$\frac{3 a^{2} \cdot 12 - 3 a b \cdot 3 - a b \cdot 4}{12} + \frac{b^{2}}{12}$

Step 3.9

Combine the numerators over the common denominator.

$\frac{3 a^{2} \cdot 12 - 3 a b \cdot 3 - a b \cdot 4 + b^{2}}{12}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Multiply $12$ by $3$ .

$\frac{36 a^{2} - 3 a b \cdot 3 - a b \cdot 4 + b^{2}}{12}$

Step 4.2

Multiply $3$ by $- 3$ .

$\frac{36 a^{2} - 9 a b - a b \cdot 4 + b^{2}}{12}$

Step 4.3

Multiply $4$ by $- 1$ .

$\frac{36 a^{2} - 9 a b - 4 a b + b^{2}}{12}$

Step 4.4

Subtract $4 a b$ from $- 9 a b$ .

$\frac{36 a^{2} - 13 a b + b^{2}}{12}$

Step 4.5

Factor by grouping.

Tap for more steps...

Step 4.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 36 \cdot 1 = 36$ and whose sum is $b = - 13$ .

Tap for more steps...

Step 4.5.1.1

Reorder terms.

$\frac{36 a^{2} + b^{2} - 13 a b}{12}$

Step 4.5.1.2

Reorder $b^{2}$ and $- 13 a b$ .

$\frac{36 a^{2} - 13 a b + b^{2}}{12}$

Step 4.5.1.3

Factor $- 13$ out of $- 13 a b$ .

$\frac{36 a^{2} - 13 (a b) + b^{2}}{12}$

Step 4.5.1.4

Rewrite $- 13$ as $- 4$ plus $- 9$

$\frac{36 a^{2} + (- 4 - 9) (a b) + b^{2}}{12}$

Step 4.5.1.5

Apply the distributive property.

$\frac{36 a^{2} - 4 (a b) - 9 (a b) + b^{2}}{12}$

Step 4.5.1.6

Move parentheses.

$\frac{36 a^{2} - 4 a b - 9 a b + b^{2}}{12}$

Step 4.5.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.5.2.1

Group the first two terms and the last two terms.

$\frac{(36 a^{2} - 4 a b) - 9 a b + b^{2}}{12}$

Step 4.5.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{4 a (9 a - b) - b (9 a - b)}{12}$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $9 a - b$ .

$\frac{(9 a - b) (4 a - b)}{12}$