Calculus Examples

$4 (x - 1) (x + 3)$

Step 1

Simplify by multiplying through.

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Step 1.1

Apply the distributive property.

$f (x) = (4 x + 4 \cdot - 1) (x + 3)$

Step 1.2

Multiply $4$ by $- 1$ .

$f (x) = (4 x - 4) (x + 3)$

Step 2

Expand $(4 x - 4) (x + 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$f (x) = 4 x (x + 3) - 4 (x + 3)$

Step 2.2

Apply the distributive property.

$f (x) = 4 x \cdot x + 4 x \cdot 3 - 4 (x + 3)$

Step 2.3

Apply the distributive property.

$f (x) = 4 x \cdot x + 4 x \cdot 3 - 4 x - 4 \cdot 3$

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 $x$ by $x$ by adding the exponents.

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Step 3.1.1.1

Move $x$ .

$f (x) = 4 (x \cdot x) + 4 x \cdot 3 - 4 x - 4 \cdot 3$

Step 3.1.1.2

Multiply $x$ by $x$ .

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

Step 3.1.2

Multiply $3$ by $4$ .

$f (x) = 4 x^{2} + 12 x - 4 x - 4 \cdot 3$

Step 3.1.3

Multiply $- 4$ by $3$ .

$f (x) = 4 x^{2} + 12 x - 4 x - 12$

Step 3.2

Subtract $4 x$ from $12 x$ .

$f (x) = 4 x^{2} + 8 x - 12$

Step 4

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 5

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

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Step 5.1

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

$x = - \frac{8}{2 (4)}$

Step 5.2

Remove parentheses.

$x = - \frac{8}{2 (4)}$

Step 5.3

Simplify $- \frac{8}{2 (4)}$ .

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Step 5.3.1

Cancel the common factor of $8$ and $2$ .

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Step 5.3.1.1

Factor $2$ out of $8$ .

$x = - \frac{2 \cdot 4}{2 \cdot 4}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $2$ out of $2 \cdot 4$ .

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

Step 5.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 4}{2 \cdot 4}$

Step 5.3.1.2.3

Rewrite the expression.

$x = - \frac{4}{4}$

Step 5.3.2

Cancel the common factor of $4$ .

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Step 5.3.2.1

Cancel the common factor.

$x = - \frac{4}{4}$

Step 5.3.2.2

Rewrite the expression.

$x = - 1 \cdot 1$

Step 5.3.3

Multiply $- 1$ by $1$ .

$x = - 1$

Step 6

Evaluate $f (- 1)$ .

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Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = 4 {(- 1)}^{2} + 8 (- 1) - 12$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

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

$f (- 1) = 4 \cdot 1 + 8 (- 1) - 12$

Step 6.2.1.2

Multiply $4$ by $1$ .

$f (- 1) = 4 + 8 (- 1) - 12$

Step 6.2.1.3

Multiply $8$ by $- 1$ .

$f (- 1) = 4 - 8 - 12$

Step 6.2.2

Simplify by subtracting numbers.

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Step 6.2.2.1

Subtract $8$ from $4$ .

$f (- 1) = - 4 - 12$

Step 6.2.2.2

Subtract $12$ from $- 4$ .

$f (- 1) = - 16$

Step 6.2.3

The final answer is $- 16$ .

$- 16$

Step 7

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

$(- 1, - 16)$

Step 8