Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

Multiply $- 1$ by $- 3$ .

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

Step 2.1.3.2

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Move $2$ to the left of $x^{2}$ .

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

Step 3.2

Multiply $2$ by $- 3$ .

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

Step 3.3

Subtract $6 x$ from $- x$ .

$\frac{2 x^{2} - 7 x + 3}{2}$

Step 3.4

Factor by grouping.

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Step 3.4.1

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 $a \cdot c = 2 \cdot 3 = 6$ and whose sum is $b = - 7$ .

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Step 3.4.1.1

Factor $- 7$ out of $- 7 x$ .

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

Step 3.4.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

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

Step 3.4.1.3

Apply the distributive property.

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

Step 3.4.2

Factor out the greatest common factor from each group.

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Step 3.4.2.1

Group the first two terms and the last two terms.

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

Step 3.4.2.2

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

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

Step 3.4.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

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