Algebra Examples

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

Step 1

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

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

Step 2

Expand $(\frac{x}{2} - 3) (x + \frac{4}{3})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\frac{x}{2} x$ .

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Step 3.1.1.1

Combine $\frac{x}{2}$ and $x$ .

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

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

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Factor $2$ out of $4$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.1.3

Combine $x$ and $\frac{2}{3}$ .

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

Step 3.1.4

Move $2$ to the left of $x$ .

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

Step 3.1.5

Cancel the common factor of $3$ .

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Step 3.1.5.1

Factor $3$ out of $- 3$ .

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

Step 3.1.5.2

Cancel the common factor.

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

Step 3.1.5.3

Rewrite the expression.

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

Step 3.1.6

Multiply $- 1$ by $4$ .

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

Step 3.2

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

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

Step 3.3

Combine $- 3 x$ and $\frac{3}{3}$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

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

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

Step 3.6

To write $\frac{2 x - 3 x \cdot 3}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.7.1

Multiply $\frac{x^{2}}{2}$ by $\frac{3}{3}$ .

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

Step 3.7.2

Multiply $2$ by $3$ .

$\frac{x^{2} \cdot 3}{6} + \frac{2 x - 3 x \cdot 3}{3} \cdot \frac{2}{2} - 4$

Step 3.7.3

Multiply $\frac{2 x - 3 x \cdot 3}{3}$ by $\frac{2}{2}$ .

$\frac{x^{2} \cdot 3}{6} + \frac{(2 x - 3 x \cdot 3) \cdot 2}{3 \cdot 2} - 4$

Step 3.7.4

Multiply $3$ by $2$ .

$\frac{x^{2} \cdot 3}{6} + \frac{(2 x - 3 x \cdot 3) \cdot 2}{6} - 4$

Step 3.8

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 3 + (2 x - 3 x \cdot 3) \cdot 2}{6} - 4$

Step 3.9

Reorder terms.

$\frac{3 x^{2} + 2 (2 x - 3 x \cdot 3)}{6} - 4$

Step 3.10

Combine $\frac{3 x^{2} + 2 (2 x - 3 x \cdot 3)}{6}$ and $- 4$ using a common denominator.

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Step 3.10.1

Move $x$ .

$\frac{3 x^{2} + 2 (2 x - 3 \cdot 3 x)}{6} - 4$

Step 3.10.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{3 x^{2} + 2 (2 x - 3 \cdot 3 x)}{6} - 4 \cdot \frac{6}{6}$

Step 3.10.3

Combine $- 4$ and $\frac{6}{6}$ .

$\frac{3 x^{2} + 2 (2 x - 3 \cdot 3 x)}{6} + \frac{- 4 \cdot 6}{6}$

Step 3.10.4

Combine the numerators over the common denominator.

$\frac{3 x^{2} + 2 (2 x - 3 \cdot 3 x) - 4 \cdot 6}{6}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $- 3$ by $3$ .

$\frac{3 x^{2} + 2 (2 x - 9 x) - 4 \cdot 6}{6}$

Step 4.2

Subtract $9 x$ from $2 x$ .

$\frac{3 x^{2} + 2 (- 7 x) - 4 \cdot 6}{6}$

Step 4.3

Multiply $- 7$ by $2$ .

$\frac{3 x^{2} - 14 x - 4 \cdot 6}{6}$

Step 4.4

Multiply $- 4$ by $6$ .

$\frac{3 x^{2} - 14 x - 24}{6}$

Step 4.5

Factor by grouping.

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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 $a \cdot c = 3 \cdot - 24 = - 72$ and whose sum is $b = - 14$ .

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Step 4.5.1.1

Factor $- 14$ out of $- 14 x$ .

$\frac{3 x^{2} - 14 (x) - 24}{6}$

Step 4.5.1.2

Rewrite $- 14$ as $4$ plus $- 18$

$\frac{3 x^{2} + (4 - 18) x - 24}{6}$

Step 4.5.1.3

Apply the distributive property.

$\frac{3 x^{2} + 4 x - 18 x - 24}{6}$

Step 4.5.2

Factor out the greatest common factor from each group.

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Step 4.5.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 4 x) - 18 x - 24}{6}$

Step 4.5.2.2

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

$\frac{x (3 x + 4) - 6 (3 x + 4)}{6}$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

$\frac{(3 x + 4) (x - 6)}{6}$