Precalculus Examples

$(x + 3) (x - 2) = 0$

Step 1

Expand $(x + 3) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

$f (x) = x (x - 2) + 3 (x - 2)$

Step 1.2

Apply the distributive property.

$f (x) = x \cdot x + x \cdot - 2 + 3 (x - 2)$

Step 1.3

Apply the distributive property.

$f (x) = x \cdot x + x \cdot - 2 + 3 x + 3 \cdot - 2$

Step 2

Simplify and combine like terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $x$ by $x$ .

$f (x) = x^{2} + x \cdot - 2 + 3 x + 3 \cdot - 2$

Step 2.1.2

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

$f (x) = x^{2} - 2 \cdot x + 3 x + 3 \cdot - 2$

Step 2.1.3

Multiply $3$ by $- 2$ .

$f (x) = x^{2} - 2 x + 3 x - 6$

Step 2.2

Add $- 2 x$ and $3 x$ .

$f (x) = x^{2} + x - 6$

Step 3

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 4

Find the value of $x = - \frac{b}{2 a}$ .

Tap for more steps...

Step 4.1

Substitute in the values of $a$ and $b$ .

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

Step 4.2

Remove parentheses.

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

Step 4.3

Cancel the common factor of $1$ .

Tap for more steps...

Step 4.3.1

Cancel the common factor.

$x = - \frac{1}{2 \cdot 1}$

Step 4.3.2

Rewrite the expression.

$x = - \frac{1}{2}$

Step 5

Evaluate $f (- \frac{1}{2})$ .

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

$f (- \frac{1}{2}) = {(- \frac{1}{2})}^{2} - \frac{1}{2} - 6$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Remove parentheses.

$f (- \frac{1}{2}) = {(- \frac{1}{2})}^{2} - \frac{1}{2} - 6$

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 5.2.2.1.1

Apply the product rule to $- \frac{1}{2}$ .

$f (- \frac{1}{2}) = {(- 1)}^{2} {(\frac{1}{2})}^{2} - \frac{1}{2} - 6$

Step 5.2.2.1.2

Apply the product rule to $\frac{1}{2}$ .

$f (- \frac{1}{2}) = {(- 1)}^{2} (\frac{1^{2}}{2^{2}}) - \frac{1}{2} - 6$

Step 5.2.2.2

Raise $- 1$ to the power of $2$ .

$f (- \frac{1}{2}) = 1 (\frac{1^{2}}{2^{2}}) - \frac{1}{2} - 6$

Step 5.2.2.3

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$f (- \frac{1}{2}) = \frac{1^{2}}{2^{2}} - \frac{1}{2} - 6$

Step 5.2.2.4

One to any power is one.

$f (- \frac{1}{2}) = \frac{1}{2^{2}} - \frac{1}{2} - 6$

Step 5.2.2.5

Raise $2$ to the power of $2$ .

$f (- \frac{1}{2}) = \frac{1}{4} - \frac{1}{2} - 6$

Step 5.2.3

Find the common denominator.

Tap for more steps...

Step 5.2.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$f (- \frac{1}{2}) = \frac{1}{4} - (\frac{1}{2} \cdot \frac{2}{2}) - 6$

Step 5.2.3.2

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$f (- \frac{1}{2}) = \frac{1}{4} - \frac{2}{2 \cdot 2} - 6$

Step 5.2.3.3

Write $- 6$ as a fraction with denominator $1$ .

$f (- \frac{1}{2}) = \frac{1}{4} - \frac{2}{2 \cdot 2} + \frac{- 6}{1}$

Step 5.2.3.4

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$f (- \frac{1}{2}) = \frac{1}{4} - \frac{2}{2 \cdot 2} + \frac{- 6}{1} \cdot \frac{4}{4}$

Step 5.2.3.5

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$f (- \frac{1}{2}) = \frac{1}{4} - \frac{2}{2 \cdot 2} + \frac{- 6 \cdot 4}{4}$

Step 5.2.3.6

Multiply $2$ by $2$ .

$f (- \frac{1}{2}) = \frac{1}{4} - \frac{2}{4} + \frac{- 6 \cdot 4}{4}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (- \frac{1}{2}) = \frac{1 - 2 - 6 \cdot 4}{4}$

Step 5.2.5

Simplify the expression.

Tap for more steps...

Step 5.2.5.1

Multiply $- 6$ by $4$ .

$f (- \frac{1}{2}) = \frac{1 - 2 - 24}{4}$

Step 5.2.5.2

Subtract $2$ from $1$ .

$f (- \frac{1}{2}) = \frac{- 1 - 24}{4}$

Step 5.2.5.3

Subtract $24$ from $- 1$ .

$f (- \frac{1}{2}) = \frac{- 25}{4}$

Step 5.2.5.4

Move the negative in front of the fraction.

$f (- \frac{1}{2}) = - \frac{25}{4}$

Step 5.2.6

The final answer is $- \frac{25}{4}$ .

$- \frac{25}{4}$

Step 6

Use the $x$ and $y$ values to find where the minimum occurs.

$(- \frac{1}{2}, - \frac{25}{4})$

Step 7