Algebra Examples

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

Step 1

Combine $\frac{3}{5}$ and $x$ .

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

Step 2

Expand $(\frac{3 x}{5} - 2) (2 x - \frac{1}{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

Multiply $2 \frac{3 x}{5}$ .

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Step 3.1.2.1

Combine $2$ and $\frac{3 x}{5}$ .

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

Step 3.1.2.2

Multiply $3$ by $2$ .

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

Step 3.1.3

Multiply $\frac{6 x}{5} x$ .

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Step 3.1.3.1

Combine $\frac{6 x}{5}$ and $x$ .

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

Add $1$ and $1$ .

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

Step 3.1.4

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

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Step 3.1.4.1

Multiply $\frac{3 x}{5}$ by $\frac{1}{2}$ .

$\frac{6 x^{2}}{5} - \frac{3 x}{5 \cdot 2} - 2 (2 x) - 2 (- \frac{1}{2})$

Step 3.1.4.2

Multiply $5$ by $2$ .

$\frac{6 x^{2}}{5} - \frac{3 x}{10} - 2 (2 x) - 2 (- \frac{1}{2})$

Step 3.1.5

Multiply $2$ by $- 2$ .

$\frac{6 x^{2}}{5} - \frac{3 x}{10} - 4 x - 2 (- \frac{1}{2})$

Step 3.1.6

Cancel the common factor of $2$ .

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Step 3.1.6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{6 x^{2}}{5} - \frac{3 x}{10} - 4 x - 2 (\frac{- 1}{2})$

Step 3.1.6.2

Factor $2$ out of $- 2$ .

$\frac{6 x^{2}}{5} - \frac{3 x}{10} - 4 x + 2 (- 1) \frac{- 1}{2}$

Step 3.1.6.3

Cancel the common factor.

$\frac{6 x^{2}}{5} - \frac{3 x}{10} - 4 x + 2 \cdot - 1 \frac{- 1}{2}$

Step 3.1.6.4

Rewrite the expression.

$\frac{6 x^{2}}{5} - \frac{3 x}{10} - 4 x - 1 \cdot - 1$

Step 3.1.7

Multiply $- 1$ by $- 1$ .

$\frac{6 x^{2}}{5} - \frac{3 x}{10} - 4 x + 1$

Step 3.2

To write $- 4 x$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{6 x^{2}}{5} - \frac{3 x}{10} - 4 x \cdot \frac{10}{10} + 1$

Step 3.3

Combine $- 4 x$ and $\frac{10}{10}$ .

$\frac{6 x^{2}}{5} - \frac{3 x}{10} + \frac{- 4 x \cdot 10}{10} + 1$

Step 3.4

Combine the numerators over the common denominator.

$\frac{6 x^{2}}{5} + \frac{- 3 x - 4 x \cdot 10}{10} + 1$

Step 3.5

To write $\frac{6 x^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{6 x^{2}}{5} \cdot \frac{2}{2} + \frac{- 3 x - 4 x \cdot 10}{10} + 1$

Step 3.6

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.6.1

Multiply $\frac{6 x^{2}}{5}$ by $\frac{2}{2}$ .

$\frac{6 x^{2} \cdot 2}{5 \cdot 2} + \frac{- 3 x - 4 x \cdot 10}{10} + 1$

Step 3.6.2

Multiply $5$ by $2$ .

$\frac{6 x^{2} \cdot 2}{10} + \frac{- 3 x - 4 x \cdot 10}{10} + 1$

Step 3.7

Combine the numerators over the common denominator.

$\frac{6 x^{2} \cdot 2 - 3 x - 4 x \cdot 10}{10} + 1$

Step 3.8

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

$\frac{6 x^{2} \cdot 2 - 3 x - 4 x \cdot 10}{10} + \frac{10}{10}$

Step 3.9

Combine the numerators over the common denominator.

$\frac{6 x^{2} \cdot 2 - 3 x - 4 x \cdot 10 + 10}{10}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $2$ by $6$ .

$\frac{12 x^{2} - 3 x - 4 x \cdot 10 + 10}{10}$

Step 4.2

Multiply $10$ by $- 4$ .

$\frac{12 x^{2} - 3 x - 40 x + 10}{10}$

Step 4.3

Subtract $40 x$ from $- 3 x$ .

$\frac{12 x^{2} - 43 x + 10}{10}$

Step 4.4

Factor by grouping.

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Step 4.4.1

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 $a \cdot c = 12 \cdot 10 = 120$ and whose sum is $b = - 43$ .

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Step 4.4.1.1

Factor $- 43$ out of $- 43 x$ .

$\frac{12 x^{2} - 43 (x) + 10}{10}$

Step 4.4.1.2

Rewrite $- 43$ as $- 3$ plus $- 40$

$\frac{12 x^{2} + (- 3 - 40) x + 10}{10}$

Step 4.4.1.3

Apply the distributive property.

$\frac{12 x^{2} - 3 x - 40 x + 10}{10}$

Step 4.4.2

Factor out the greatest common factor from each group.

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Step 4.4.2.1

Group the first two terms and the last two terms.

$\frac{(12 x^{2} - 3 x) - 40 x + 10}{10}$

Step 4.4.2.2

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

$\frac{3 x (4 x - 1) - 10 (4 x - 1)}{10}$

Step 4.4.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 1$ .

$\frac{(4 x - 1) (3 x - 10)}{10}$