Examples

f (x) = x^{2} - 5 x + 6

$f (x) = x^{2} - 5 x + 6$

Step 1

The minimum of a quadratic function occurs at

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$ . If

a

$a$ is positive, the minimum value of the function is

f (- \frac{b}{2 a})

$f (- \frac{b}{2 a})$ .

f_{min}

$f_{min}$

x = a x^{2} + b x + c

$x = a x^{2} + b x + c$ occurs at

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$

Step 2

Find the value of

x = - \frac{b}{2 a}

$x = - \frac{b}{2 a}$ .

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Step 2.1

Substitute in the values of

a

$a$ and

b

$b$ .

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

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

Step 2.2

Remove parentheses.

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

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

Step 2.3

Simplify

- \frac{- 5}{2 (1)}

$- \frac{- 5}{2 (1)}$ .

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Step 2.3.1

Multiply

2

$2$ by

1

$1$ .

x = - \frac{- 5}{2}

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

Step 2.3.2

Move the negative in front of the fraction.

x = - - \frac{5}{2}

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

Step 2.3.3

Multiply

- - \frac{5}{2}

$- - \frac{5}{2}$ .

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Step 2.3.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

x = 1 (\frac{5}{2})

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

Step 2.3.3.2

Multiply

\frac{5}{2}

$\frac{5}{2}$ by

1

$1$ .

x = \frac{5}{2}

$x = \frac{5}{2}$

x = \frac{5}{2}

$x = \frac{5}{2}$

x = \frac{5}{2}

$x = \frac{5}{2}$

x = \frac{5}{2}

$x = \frac{5}{2}$

Step 3

Evaluate

f (\frac{5}{2})

$f (\frac{5}{2})$ .

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Step 3.1

Replace the variable

x

$x$ with

\frac{5}{2}

$\frac{5}{2}$ in the expression.

f (\frac{5}{2}) = {(\frac{5}{2})}^{2} - 5 (\frac{5}{2}) + 6

$f (\frac{5}{2}) = {(\frac{5}{2})}^{2} - 5 (\frac{5}{2}) + 6$

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

Apply the product rule to

\frac{5}{2}

$\frac{5}{2}$ .

f (\frac{5}{2}) = \frac{5^{2}}{2^{2}} - 5 (\frac{5}{2}) + 6

$f (\frac{5}{2}) = \frac{5^{2}}{2^{2}} - 5 (\frac{5}{2}) + 6$

Step 3.2.1.2

Raise

5

$5$ to the power of

2

$2$ .

f (\frac{5}{2}) = \frac{25}{2^{2}} - 5 (\frac{5}{2}) + 6

$f (\frac{5}{2}) = \frac{25}{2^{2}} - 5 (\frac{5}{2}) + 6$

Step 3.2.1.3

Raise

2

$2$ to the power of

2

$2$ .

f (\frac{5}{2}) = \frac{25}{4} - 5 (\frac{5}{2}) + 6

$f (\frac{5}{2}) = \frac{25}{4} - 5 (\frac{5}{2}) + 6$

Step 3.2.1.4

Multiply

- 5 (\frac{5}{2})

$- 5 (\frac{5}{2})$ .

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Step 3.2.1.4.1

Combine

- 5

$- 5$ and

\frac{5}{2}

$\frac{5}{2}$ .

f (\frac{5}{2}) = \frac{25}{4} + \frac{- 5 \cdot 5}{2} + 6

$f (\frac{5}{2}) = \frac{25}{4} + \frac{- 5 \cdot 5}{2} + 6$

Step 3.2.1.4.2

Multiply

- 5

$- 5$ by

5

$5$ .

f (\frac{5}{2}) = \frac{25}{4} + \frac{- 25}{2} + 6

$f (\frac{5}{2}) = \frac{25}{4} + \frac{- 25}{2} + 6$

f (\frac{5}{2}) = \frac{25}{4} + \frac{- 25}{2} + 6

$f (\frac{5}{2}) = \frac{25}{4} + \frac{- 25}{2} + 6$

Step 3.2.1.5

Move the negative in front of the fraction.

f (\frac{5}{2}) = \frac{25}{4} - \frac{25}{2} + 6

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25}{2} + 6$

f (\frac{5}{2}) = \frac{25}{4} - \frac{25}{2} + 6

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25}{2} + 6$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply

\frac{25}{2}

$\frac{25}{2}$ by

\frac{2}{2}

$\frac{2}{2}$ .

f (\frac{5}{2}) = \frac{25}{4} - (\frac{25}{2} \cdot \frac{2}{2}) + 6

$f (\frac{5}{2}) = \frac{25}{4} - (\frac{25}{2} \cdot \frac{2}{2}) + 6$

Step 3.2.2.2

Multiply

\frac{25}{2}

$\frac{25}{2}$ by

\frac{2}{2}

$\frac{2}{2}$ .

f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + 6

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + 6$

Step 3.2.2.3

Write

6

$6$ as a fraction with denominator

1

$1$ .

f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6}{1}

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6}{1}$

Step 3.2.2.4

Multiply

\frac{6}{1}

$\frac{6}{1}$ by

\frac{4}{4}

$\frac{4}{4}$ .

f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6}{1} \cdot \frac{4}{4}

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6}{1} \cdot \frac{4}{4}$

Step 3.2.2.5

Multiply

\frac{6}{1}

$\frac{6}{1}$ by

\frac{4}{4}

$\frac{4}{4}$ .

f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6 \cdot 4}{4}

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{2 \cdot 2} + \frac{6 \cdot 4}{4}$

Step 3.2.2.6

Multiply

2

$2$ by

2

$2$ .

f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{4} + \frac{6 \cdot 4}{4}

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{4} + \frac{6 \cdot 4}{4}$

f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{4} + \frac{6 \cdot 4}{4}

$f (\frac{5}{2}) = \frac{25}{4} - \frac{25 \cdot 2}{4} + \frac{6 \cdot 4}{4}$

Step 3.2.3

Combine the numerators over the common denominator.

f (\frac{5}{2}) = \frac{25 - 25 \cdot 2 + 6 \cdot 4}{4}

$f (\frac{5}{2}) = \frac{25 - 25 \cdot 2 + 6 \cdot 4}{4}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply

- 25

$- 25$ by

2

$2$ .

f (\frac{5}{2}) = \frac{25 - 50 + 6 \cdot 4}{4}

$f (\frac{5}{2}) = \frac{25 - 50 + 6 \cdot 4}{4}$

Step 3.2.4.2

Multiply

6

$6$ by

4

$4$ .

f (\frac{5}{2}) = \frac{25 - 50 + 24}{4}

$f (\frac{5}{2}) = \frac{25 - 50 + 24}{4}$

f (\frac{5}{2}) = \frac{25 - 50 + 24}{4}

$f (\frac{5}{2}) = \frac{25 - 50 + 24}{4}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract

50

$50$ from

25

$25$ .

f (\frac{5}{2}) = \frac{- 25 + 24}{4}

$f (\frac{5}{2}) = \frac{- 25 + 24}{4}$

Step 3.2.5.2

Add

- 25

$- 25$ and

24

$24$ .

f (\frac{5}{2}) = \frac{- 1}{4}

$f (\frac{5}{2}) = \frac{- 1}{4}$

Step 3.2.5.3

Move the negative in front of the fraction.

f (\frac{5}{2}) = - \frac{1}{4}

$f (\frac{5}{2}) = - \frac{1}{4}$

f (\frac{5}{2}) = - \frac{1}{4}

$f (\frac{5}{2}) = - \frac{1}{4}$

Step 3.2.6

The final answer is

- \frac{1}{4}

$- \frac{1}{4}$ .

- \frac{1}{4}

$- \frac{1}{4}$

- \frac{1}{4}

$- \frac{1}{4}$

- \frac{1}{4}

$- \frac{1}{4}$

Step 4

Use the

x

$x$ and

y

$y$ values to find where the minimum occurs.

(\frac{5}{2}, - \frac{1}{4})

$(\frac{5}{2}, - \frac{1}{4})$

Step 5

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