Precalculus Examples

${(3 x)}^{2} - 2 x - 5$

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $3 x$ .

$f (x) = 3^{2} x^{2} - 2 x - 5$

Step 1.2

Raise $3$ to the power of $2$ .

$f (x) = 9 x^{2} - 2 x - 5$

Step 2

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 3

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

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Step 3.1

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

$x = - \frac{- 2}{2 (9)}$

Step 3.2

Remove parentheses.

$x = - \frac{- 2}{2 (9)}$

Step 3.3

Simplify $- \frac{- 2}{2 (9)}$ .

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Step 3.3.1

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

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Step 3.3.1.1

Factor $2$ out of $- 2$ .

$x = - \frac{2 \cdot - 1}{2 \cdot 9}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $2 \cdot 9$ .

$x = - \frac{2 \cdot - 1}{2 (9)}$

Step 3.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 1}{2 \cdot 9}$

Step 3.3.1.2.3

Rewrite the expression.

$x = - \frac{- 1}{9}$

Step 3.3.2

Move the negative in front of the fraction.

$x = - - \frac{1}{9}$

Step 3.3.3

Multiply $- - \frac{1}{9}$ .

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Step 3.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{1}{9})$

Step 3.3.3.2

Multiply $\frac{1}{9}$ by $1$ .

$x = \frac{1}{9}$

Step 4

Evaluate $f (\frac{1}{9})$ .

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Step 4.1

Replace the variable $x$ with $\frac{1}{9}$ in the expression.

$f (\frac{1}{9}) = 9 {(\frac{1}{9})}^{2} - 2 (\frac{1}{9}) - 5$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the product rule to $\frac{1}{9}$ .

$f (\frac{1}{9}) = 9 (\frac{1^{2}}{9^{2}}) - 2 (\frac{1}{9}) - 5$

Step 4.2.1.2

One to any power is one.

$f (\frac{1}{9}) = 9 (\frac{1}{9^{2}}) - 2 (\frac{1}{9}) - 5$

Step 4.2.1.3

Raise $9$ to the power of $2$ .

$f (\frac{1}{9}) = 9 (\frac{1}{81}) - 2 (\frac{1}{9}) - 5$

Step 4.2.1.4

Cancel the common factor of $9$ .

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Step 4.2.1.4.1

Factor $9$ out of $81$ .

$f (\frac{1}{9}) = 9 (\frac{1}{9 (9)}) - 2 (\frac{1}{9}) - 5$

Step 4.2.1.4.2

Cancel the common factor.

$f (\frac{1}{9}) = 9 (\frac{1}{9 \cdot 9}) - 2 (\frac{1}{9}) - 5$

Step 4.2.1.4.3

Rewrite the expression.

$f (\frac{1}{9}) = \frac{1}{9} - 2 (\frac{1}{9}) - 5$

Step 4.2.1.5

Combine $- 2$ and $\frac{1}{9}$ .

$f (\frac{1}{9}) = \frac{1}{9} + \frac{- 2}{9} - 5$

Step 4.2.1.6

Move the negative in front of the fraction.

$f (\frac{1}{9}) = \frac{1}{9} - \frac{2}{9} - 5$

Step 4.2.2

Combine fractions.

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Step 4.2.2.1

Combine the numerators over the common denominator.

$f (\frac{1}{9}) = - 5 + \frac{1 - 2}{9}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Subtract $2$ from $1$ .

$f (\frac{1}{9}) = - 5 + \frac{- 1}{9}$

Step 4.2.2.2.2

Move the negative in front of the fraction.

$f (\frac{1}{9}) = - 5 - \frac{1}{9}$

Step 4.2.3

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{1}{9}) = - 5 \cdot \frac{9}{9} - \frac{1}{9}$

Step 4.2.4

Combine $- 5$ and $\frac{9}{9}$ .

$f (\frac{1}{9}) = \frac{- 5 \cdot 9}{9} - \frac{1}{9}$

Step 4.2.5

Combine the numerators over the common denominator.

$f (\frac{1}{9}) = \frac{- 5 \cdot 9 - 1}{9}$

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $- 5$ by $9$ .

$f (\frac{1}{9}) = \frac{- 45 - 1}{9}$

Step 4.2.6.2

Subtract $1$ from $- 45$ .

$f (\frac{1}{9}) = \frac{- 46}{9}$

Step 4.2.7

Move the negative in front of the fraction.

$f (\frac{1}{9}) = - \frac{46}{9}$

Step 4.2.8

The final answer is $- \frac{46}{9}$ .

$- \frac{46}{9}$

Step 5

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

$(\frac{1}{9}, - \frac{46}{9})$

Step 6