Algebra Examples

$f (x) = \frac{1}{4} {(x + 4)}^{2} + 3$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$f (x) = \frac{1}{4} \cdot ((x + 4) (x + 4)) + 3$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$f (x) = \frac{1}{4} \cdot (x (x + 4) + 4 (x + 4)) + 3$

Step 1.2.2

Apply the distributive property.

$f (x) = \frac{1}{4} \cdot (x \cdot x + x \cdot 4 + 4 (x + 4)) + 3$

Step 1.2.3

Apply the distributive property.

$f (x) = \frac{1}{4} \cdot (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4) + 3$

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 $x$ .

$f (x) = \frac{1}{4} \cdot (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4) + 3$

Step 1.3.1.2

Move $4$ to the left of $x$ .

$f (x) = \frac{1}{4} \cdot (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4) + 3$

Step 1.3.1.3

Multiply $4$ by $4$ .

$f (x) = \frac{1}{4} \cdot (x^{2} + 4 x + 4 x + 16) + 3$

Step 1.3.2

Add $4 x$ and $4 x$ .

$f (x) = \frac{1}{4} \cdot (x^{2} + 8 x + 16) + 3$

Step 1.4

Apply the distributive property.

$f (x) = \frac{1}{4} x^{2} + \frac{1}{4} \cdot (8 x) + \frac{1}{4} \cdot 16 + 3$

Step 1.5

Simplify.

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Step 1.5.1

Combine $\frac{1}{4}$ and $x^{2}$ .

$f (x) = \frac{x^{2}}{4} + \frac{1}{4} \cdot (8 x) + \frac{1}{4} \cdot 16 + 3$

Step 1.5.2

Cancel the common factor of $4$ .

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Step 1.5.2.1

Factor $4$ out of $8 x$ .

$f (x) = \frac{x^{2}}{4} + \frac{1}{4} \cdot (4 (2 x)) + \frac{1}{4} \cdot 16 + 3$

Step 1.5.2.2

Cancel the common factor.

$f (x) = \frac{x^{2}}{4} + \frac{1}{4} \cdot (4 (2 x)) + \frac{1}{4} \cdot 16 + 3$

Step 1.5.2.3

Rewrite the expression.

$f (x) = \frac{x^{2}}{4} + 2 x + \frac{1}{4} \cdot 16 + 3$

Step 1.5.3

Cancel the common factor of $4$ .

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Step 1.5.3.1

Factor $4$ out of $16$ .

$f (x) = \frac{x^{2}}{4} + 2 x + \frac{1}{4} \cdot (4 (4)) + 3$

Step 1.5.3.2

Cancel the common factor.

$f (x) = \frac{x^{2}}{4} + 2 x + \frac{1}{4} \cdot (4 \cdot 4) + 3$

Step 1.5.3.3

Rewrite the expression.

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

Step 2

Add $4$ and $3$ .

$f (x) = \frac{x^{2}}{4} + 2 x + 7$

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 $b$ .

$x = - \frac{2}{2 (0.25)}$

Step 4.2

Remove parentheses.

$x = - \frac{2}{2 (0.25)}$

Step 4.3

Simplify $- \frac{2}{2 (0.25)}$ .

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Cancel the common factor.

$x = - \frac{2}{2 \cdot 0.25}$

Step 4.3.1.2

Rewrite the expression.

$x = - \frac{1}{0.25}$

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Divide $1$ by $0.25$ .

$x = - 1 \cdot 4$

Step 4.3.2.2

Multiply $- 1$ by $4$ .

$x = - 4$

Step 5

Evaluate $f (- 4)$ .

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = \frac{{(- 4)}^{2}}{4} + 2 (- 4) + 7$

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

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 5.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$f (- 4) = \frac{{(- 1 \cdot 4)}^{2}}{4} + 2 (- 4) + 7$

Step 5.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$f (- 4) = \frac{{(- 1)}^{2} \cdot 4^{2}}{4} + 2 (- 4) + 7$

Step 5.2.1.1.3

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

$f (- 4) = \frac{1 \cdot 4^{2}}{4} + 2 (- 4) + 7$

Step 5.2.1.1.4

Multiply $4^{2}$ by $1$ .

$f (- 4) = \frac{4^{2}}{4} + 2 (- 4) + 7$

Step 5.2.1.1.5

Factor $4$ out of $4^{2}$ .

$f (- 4) = \frac{4 \cdot 4}{4} + 2 (- 4) + 7$

Step 5.2.1.1.6

Cancel the common factors.

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Step 5.2.1.1.6.1

Factor $4$ out of $4$ .

$f (- 4) = \frac{4 \cdot 4}{4 (1)} + 2 (- 4) + 7$

Step 5.2.1.1.6.2

Cancel the common factor.

$f (- 4) = \frac{4 \cdot 4}{4 \cdot 1} + 2 (- 4) + 7$

Step 5.2.1.1.6.3

Rewrite the expression.

$f (- 4) = \frac{4}{1} + 2 (- 4) + 7$

Step 5.2.1.1.6.4

Divide $4$ by $1$ .

$f (- 4) = 4 + 2 (- 4) + 7$

Step 5.2.1.2

Multiply $2$ by $- 4$ .

$f (- 4) = 4 - 8 + 7$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $8$ from $4$ .

$f (- 4) = - 4 + 7$

Step 5.2.2.2

Add $- 4$ and $7$ .

$f (- 4) = 3$

Step 5.2.3

The final answer is $3$ .

$3$

Step 6

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

$(- 4, 3)$

Step 7