Calculus Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 2.1.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

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

Step 2.1.2

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

$f (x) = x \cdot x + 0 + 3 \cdot - 3$

Step 2.1.3

Add $x \cdot x$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x$ by $x$ .

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

Step 2.2.2

Multiply $3$ by $- 3$ .

$f (x) = x^{2} - 9$

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 $x = - \frac{b}{2 a}$ .

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Step 4.1

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

$x = - \frac{0}{2 (1)}$

Step 4.2

Remove parentheses.

$x = - \frac{0}{2 (1)}$

Step 4.3

Simplify $- \frac{0}{2 (1)}$ .

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Step 4.3.1

Cancel the common factor of $0$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $0$ .

$x = - \frac{2 (0)}{2 (1)}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Cancel the common factor.

$x = - \frac{2 \cdot 0}{2 \cdot 1}$

Step 4.3.1.2.2

Rewrite the expression.

$x = - \frac{0}{1}$

Step 4.3.1.2.3

Divide $0$ by $1$ .

$x = - 0$

Step 4.3.2

Multiply $- 1$ by $0$ .

$x = 0$

Step 5

Evaluate $f (0)$ .

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2} - 9$

Step 5.2

Simplify the result.

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Step 5.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 - 9$

Step 5.2.2

Subtract $9$ from $0$ .

$f (0) = - 9$

Step 5.2.3

The final answer is $- 9$ .

$- 9$

Step 6

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

$(0, - 9)$

Step 7