Precalculus Examples

$p (x) = {(x + 5)}^{2} - 4$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$p (x) = (x + 5) (x + 5) - 4$

Step 1.2

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$p (x) = x (x + 5) + 5 (x + 5) - 4$

Step 1.2.2

Apply the distributive property.

$p (x) = x \cdot x + x \cdot 5 + 5 (x + 5) - 4$

Step 1.2.3

Apply the distributive property.

$p (x) = x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5 - 4$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$p (x) = x^{2} + x \cdot 5 + 5 x + 5 \cdot 5 - 4$

Step 1.3.1.2

Move $5$ to the left of $x$ .

$p (x) = x^{2} + 5 \cdot x + 5 x + 5 \cdot 5 - 4$

Step 1.3.1.3

Multiply $5$ by $5$ .

$p (x) = x^{2} + 5 x + 5 x + 25 - 4$

Step 1.3.2

Add $5 x$ and $5 x$ .

$p (x) = x^{2} + 10 x + 25 - 4$

Step 2

Subtract $4$ from $25$ .

$p (x) = x^{2} + 10 x + 21$

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 $b$ .

$x = - \frac{10}{2 (1)}$

Step 4.2

Remove parentheses.

$x = - \frac{10}{2 (1)}$

Step 4.3

Simplify $- \frac{10}{2 (1)}$ .

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Step 4.3.1

Cancel the common factor of $10$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $10$ .

$x = - \frac{2 \cdot 5}{2 \cdot 1}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 4.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 5}{2 \cdot 1}$

Step 4.3.1.2.3

Rewrite the expression.

$x = - \frac{5}{1}$

Step 4.3.1.2.4

Divide $5$ by $1$ .

$x = - 1 \cdot 5$

Step 4.3.2

Multiply $- 1$ by $5$ .

$x = - 5$

Step 5

Evaluate $f (- 5)$ .

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Step 5.1

Replace the variable $x$ with $- 5$ in the expression.

$p (- 5) = {(- 5)}^{2} + 10 (- 5) + 21$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 5$ to the power of $2$ .

$p (- 5) = 25 + 10 (- 5) + 21$

Step 5.2.1.2

Multiply $10$ by $- 5$ .

$p (- 5) = 25 - 50 + 21$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $50$ from $25$ .

$p (- 5) = - 25 + 21$

Step 5.2.2.2

Add $- 25$ and $21$ .

$p (- 5) = - 4$

Step 5.2.3

The final answer is $- 4$ .

$- 4$

Step 6

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

$(- 5, - 4)$

Step 7