Examples

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

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

$b$ .

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

Step 2.2

Remove parentheses.

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

Step 2.3

Multiply

$2$ by

$2$ .

$x = - \frac{5}{4}$

Step 3

Evaluate

$f (- \frac{5}{4})$ .

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Step 3.1

Replace the variable

$x$ with

$- \frac{5}{4}$ in the expression.

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

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

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.1.1

Apply the product rule to

$- \frac{5}{4}$ .

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

Step 3.2.1.1.2

Apply the product rule to

$\frac{5}{4}$ .

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

Step 3.2.1.2

Raise

$- 1$ to the power of

$2$ .

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

Step 3.2.1.3

Multiply

$\frac{5^{2}}{4^{2}}$ by

$1$ .

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

Step 3.2.1.4

Raise

$5$ to the power of

$2$ .

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

Step 3.2.1.5

Raise

$4$ to the power of

$2$ .

$f (- \frac{5}{4}) = 2 (\frac{25}{16}) + 5 (- \frac{5}{4}) - 6$

Step 3.2.1.6

Cancel the common factor of

$2$ .

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Step 3.2.1.6.1

Factor

$2$ out of

$16$ .

$f (- \frac{5}{4}) = 2 (\frac{25}{2 (8)}) + 5 (- \frac{5}{4}) - 6$

Step 3.2.1.6.2

Cancel the common factor.

$f (- \frac{5}{4}) = 2 (\frac{25}{2 \cdot 8}) + 5 (- \frac{5}{4}) - 6$

Step 3.2.1.6.3

Rewrite the expression.

$f (- \frac{5}{4}) = \frac{25}{8} + 5 (- \frac{5}{4}) - 6$

Step 3.2.1.7

Multiply

$5 (- \frac{5}{4})$ .

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Step 3.2.1.7.1

Multiply

$- 1$ by

$5$ .

$f (- \frac{5}{4}) = \frac{25}{8} - 5 (\frac{5}{4}) - 6$

Step 3.2.1.7.2

Combine

$- 5$ and

$\frac{5}{4}$ .

$f (- \frac{5}{4}) = \frac{25}{8} + \frac{- 5 \cdot 5}{4} - 6$

Step 3.2.1.7.3

Multiply

$- 5$ by

$5$ .

$f (- \frac{5}{4}) = \frac{25}{8} + \frac{- 25}{4} - 6$

Step 3.2.1.8

Move the negative in front of the fraction.

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25}{4} - 6$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply

$\frac{25}{4}$ by

$\frac{2}{2}$ .

$f (- \frac{5}{4}) = \frac{25}{8} - (\frac{25}{4} \cdot \frac{2}{2}) - 6$

Step 3.2.2.2

Multiply

$\frac{25}{4}$ by

$\frac{2}{2}$ .

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25 \cdot 2}{4 \cdot 2} - 6$

Step 3.2.2.3

Write

$- 6$ as a fraction with denominator

$1$ .

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25 \cdot 2}{4 \cdot 2} + \frac{- 6}{1}$

Step 3.2.2.4

Multiply

$\frac{- 6}{1}$ by

$\frac{8}{8}$ .

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25 \cdot 2}{4 \cdot 2} + \frac{- 6}{1} \cdot \frac{8}{8}$

Step 3.2.2.5

Multiply

$\frac{- 6}{1}$ by

$\frac{8}{8}$ .

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25 \cdot 2}{4 \cdot 2} + \frac{- 6 \cdot 8}{8}$

Step 3.2.2.6

Reorder the factors of

$4 \cdot 2$ .

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25 \cdot 2}{2 \cdot 4} + \frac{- 6 \cdot 8}{8}$

Step 3.2.2.7

Multiply

$2$ by

$4$ .

$f (- \frac{5}{4}) = \frac{25}{8} - \frac{25 \cdot 2}{8} + \frac{- 6 \cdot 8}{8}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{5}{4}) = \frac{25 - 25 \cdot 2 - 6 \cdot 8}{8}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply

$- 25$ by

$2$ .

$f (- \frac{5}{4}) = \frac{25 - 50 - 6 \cdot 8}{8}$

Step 3.2.4.2

Multiply

$- 6$ by

$8$ .

$f (- \frac{5}{4}) = \frac{25 - 50 - 48}{8}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract

$50$ from

$25$ .

$f (- \frac{5}{4}) = \frac{- 25 - 48}{8}$

Step 3.2.5.2

Subtract

$48$ from

$- 25$ .

$f (- \frac{5}{4}) = \frac{- 73}{8}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (- \frac{5}{4}) = - \frac{73}{8}$

Step 3.2.6

The final answer is

$- \frac{73}{8}$ .

$- \frac{73}{8}$

Step 4

Use the

$x$ and

$y$ values to find where the minimum occurs.

$(- \frac{5}{4}, - \frac{73}{8})$

Step 5

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