Finite Math Examples

$f (x) = 0.5 {(x - 6)}^{2} + 4$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$f (x) = 0.5 ((x - 6) (x - 6)) + 4$

Step 1.2

Expand $(x - 6) (x - 6)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = 0.5 (x (x - 6) - 6 (x - 6)) + 4$

Step 1.2.2

Apply the distributive property.

$f (x) = 0.5 (x \cdot x + x \cdot - 6 - 6 (x - 6)) + 4$

Step 1.2.3

Apply the distributive property.

$f (x) = 0.5 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) + 4$

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 $x$ .

$f (x) = 0.5 (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) + 4$

Step 1.3.1.2

Move $- 6$ to the left of $x$ .

$f (x) = 0.5 (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) + 4$

Step 1.3.1.3

Multiply $- 6$ by $- 6$ .

$f (x) = 0.5 (x^{2} - 6 x - 6 x + 36) + 4$

Step 1.3.2

Subtract $6 x$ from $- 6 x$ .

$f (x) = 0.5 (x^{2} - 12 x + 36) + 4$

Step 1.4

Apply the distributive property.

$f (x) = 0.5 x^{2} + 0.5 (- 12 x) + 0.5 \cdot 36 + 4$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 12$ by $0.5$ .

$f (x) = 0.5 x^{2} - 6 x + 0.5 \cdot 36 + 4$

Step 1.5.2

Multiply $0.5$ by $36$ .

$f (x) = 0.5 x^{2} - 6 x + 18 + 4$

Step 2

Add $18$ and $4$ .

$f (x) = 0.5 x^{2} - 6 x + 22$

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 $b$ .

$x = - \frac{- 6}{2 (0.5)}$

Step 4.2

Remove parentheses.

$x = - \frac{- 6}{2 (0.5)}$

Step 4.3

Simplify $- \frac{- 6}{2 (0.5)}$ .

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Step 4.3.1

Cancel the common factor of $- 6$ and $2$ .

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Step 4.3.1.1

Factor $2$ out of $- 6$ .

$x = - \frac{2 \cdot - 3}{2 \cdot 0.5}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $2 \cdot 0.5$ .

$x = - \frac{2 \cdot - 3}{2 (0.5)}$

Step 4.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 3}{2 \cdot 0.5}$

Step 4.3.1.2.3

Rewrite the expression.

$x = - \frac{- 3}{0.5}$

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Divide $- 3$ by $0.5$ .

$x = - - 6$

Step 4.3.2.2

Multiply $- 1$ by $- 6$ .

$x = 6$

Step 5

Evaluate $f (6)$ .

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Step 5.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = 0.5 {(6)}^{2} - 6 \cdot 6 + 22$

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 $6$ to the power of $2$ .

$f (6) = 0.5 \cdot 36 - 6 \cdot 6 + 22$

Step 5.2.1.2

Multiply $0.5$ by $36$ .

$f (6) = 18 - 6 \cdot 6 + 22$

Step 5.2.1.3

Multiply $- 6$ by $6$ .

$f (6) = 18 - 36 + 22$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $36$ from $18$ .

$f (6) = - 18 + 22$

Step 5.2.2.2

Add $- 18$ and $22$ .

$f (6) = 4$

Step 5.2.3

The final answer is $4$ .

$4$

Step 6

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

$(6, 4)$

Step 7