Precalculus Examples

$g (x) = 18 x^{2} - 36 x + 10$

Step 1

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 2

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

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Step 2.1

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

$x = - \frac{- 36}{2 (18)}$

Step 2.2

Remove parentheses.

$x = - \frac{- 36}{2 (18)}$

Step 2.3

Simplify $- \frac{- 36}{2 (18)}$ .

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Step 2.3.1

Cancel the common factor of $- 36$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $- 36$ .

$x = - \frac{2 \cdot - 18}{2 \cdot 18}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $2 \cdot 18$ .

$x = - \frac{2 \cdot - 18}{2 (18)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 18}{2 \cdot 18}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{- 18}{18}$

Step 2.3.2

Cancel the common factor of $- 18$ and $18$ .

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Step 2.3.2.1

Factor $18$ out of $- 18$ .

$x = - \frac{18 \cdot - 1}{18}$

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor $18$ out of $18$ .

$x = - \frac{18 \cdot - 1}{18 (1)}$

Step 2.3.2.2.2

Cancel the common factor.

$x = - \frac{18 \cdot - 1}{18 \cdot 1}$

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide $- 1$ by $1$ .

$x = - - 1$

Step 2.3.3

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 3

Evaluate $f (1)$ .

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Step 3.1

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

$g (1) = 18 {(1)}^{2} - 36 \cdot 1 + 10$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

One to any power is one.

$g (1) = 18 \cdot 1 - 36 \cdot 1 + 10$

Step 3.2.1.2

Multiply $18$ by $1$ .

$g (1) = 18 - 36 \cdot 1 + 10$

Step 3.2.1.3

Multiply $- 36$ by $1$ .

$g (1) = 18 - 36 + 10$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $36$ from $18$ .

$g (1) = - 18 + 10$

Step 3.2.2.2

Add $- 18$ and $10$ .

$g (1) = - 8$

Step 3.2.3

The final answer is $- 8$ .

$- 8$

Step 4

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

$(1, - 8)$

Step 5