Examples

f (x) = 5 x^{2} + 3 x - 7

$f (x) = 5 x^{2} + 3 x - 7$

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

$b$ .

$x = - \frac{3}{2 (5)}$

Step 2.2

Remove parentheses.

$x = - \frac{3}{2 (5)}$

Step 2.3

Multiply

$2$ by

$5$ .

$x = - \frac{3}{10}$

Step 3

Evaluate

$f (- \frac{3}{10})$ .

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Step 3.1

Replace the variable

$x$ with

$- \frac{3}{10}$ in the expression.

$f (- \frac{3}{10}) = 5 {(- \frac{3}{10})}^{2} + 3 (- \frac{3}{10}) - 7$

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

$- \frac{3}{10}$ .

$f (- \frac{3}{10}) = 5 ({(- 1)}^{2} {(\frac{3}{10})}^{2}) + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.1.2

Apply the product rule to

$\frac{3}{10}$ .

$f (- \frac{3}{10}) = 5 ({(- 1)}^{2} (\frac{3^{2}}{10^{2}})) + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.2

Raise

$- 1$ to the power of

$2$ .

$f (- \frac{3}{10}) = 5 (1 (\frac{3^{2}}{10^{2}})) + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.3

Multiply

$\frac{3^{2}}{10^{2}}$ by

$1$ .

$f (- \frac{3}{10}) = 5 (\frac{3^{2}}{10^{2}}) + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.4

Raise

$3$ to the power of

$2$ .

$f (- \frac{3}{10}) = 5 (\frac{9}{10^{2}}) + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.5

Raise

$10$ to the power of

$2$ .

$f (- \frac{3}{10}) = 5 (\frac{9}{100}) + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.6

Cancel the common factor of

$5$ .

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Step 3.2.1.6.1

Factor

$5$ out of

$100$ .

$f (- \frac{3}{10}) = 5 (\frac{9}{5 (20)}) + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.6.2

Cancel the common factor.

$f (- \frac{3}{10}) = 5 (\frac{9}{5 \cdot 20}) + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.6.3

Rewrite the expression.

$f (- \frac{3}{10}) = \frac{9}{20} + 3 (- \frac{3}{10}) - 7$

Step 3.2.1.7

Multiply

$3 (- \frac{3}{10})$ .

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Step 3.2.1.7.1

Multiply

$- 1$ by

$3$ .

$f (- \frac{3}{10}) = \frac{9}{20} - 3 (\frac{3}{10}) - 7$

Step 3.2.1.7.2

Combine

$- 3$ and

$\frac{3}{10}$ .

$f (- \frac{3}{10}) = \frac{9}{20} + \frac{- 3 \cdot 3}{10} - 7$

Step 3.2.1.7.3

Multiply

$- 3$ by

$3$ .

$f (- \frac{3}{10}) = \frac{9}{20} + \frac{- 9}{10} - 7$

Step 3.2.1.8

Move the negative in front of the fraction.

$f (- \frac{3}{10}) = \frac{9}{20} - \frac{9}{10} - 7$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply

$\frac{9}{10}$ by

$\frac{2}{2}$ .

$f (- \frac{3}{10}) = \frac{9}{20} - (\frac{9}{10} \cdot \frac{2}{2}) - 7$

Step 3.2.2.2

Multiply

$\frac{9}{10}$ by

$\frac{2}{2}$ .

$f (- \frac{3}{10}) = \frac{9}{20} - \frac{9 \cdot 2}{10 \cdot 2} - 7$

Step 3.2.2.3

Write

$- 7$ as a fraction with denominator

$1$ .

$f (- \frac{3}{10}) = \frac{9}{20} - \frac{9 \cdot 2}{10 \cdot 2} + \frac{- 7}{1}$

Step 3.2.2.4

Multiply

$\frac{- 7}{1}$ by

$\frac{20}{20}$ .

$f (- \frac{3}{10}) = \frac{9}{20} - \frac{9 \cdot 2}{10 \cdot 2} + \frac{- 7}{1} \cdot \frac{20}{20}$

Step 3.2.2.5

Multiply

$\frac{- 7}{1}$ by

$\frac{20}{20}$ .

$f (- \frac{3}{10}) = \frac{9}{20} - \frac{9 \cdot 2}{10 \cdot 2} + \frac{- 7 \cdot 20}{20}$

Step 3.2.2.6

Reorder the factors of

$10 \cdot 2$ .

$f (- \frac{3}{10}) = \frac{9}{20} - \frac{9 \cdot 2}{2 \cdot 10} + \frac{- 7 \cdot 20}{20}$

Step 3.2.2.7

Multiply

$2$ by

$10$ .

$f (- \frac{3}{10}) = \frac{9}{20} - \frac{9 \cdot 2}{20} + \frac{- 7 \cdot 20}{20}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{3}{10}) = \frac{9 - 9 \cdot 2 - 7 \cdot 20}{20}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply

$- 9$ by

$2$ .

$f (- \frac{3}{10}) = \frac{9 - 18 - 7 \cdot 20}{20}$

Step 3.2.4.2

Multiply

$- 7$ by

$20$ .

$f (- \frac{3}{10}) = \frac{9 - 18 - 140}{20}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract

$18$ from

$9$ .

$f (- \frac{3}{10}) = \frac{- 9 - 140}{20}$

Step 3.2.5.2

Subtract

$140$ from

$- 9$ .

$f (- \frac{3}{10}) = \frac{- 149}{20}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (- \frac{3}{10}) = - \frac{149}{20}$

Step 3.2.6

The final answer is

$- \frac{149}{20}$ .

$- \frac{149}{20}$

Step 4

Use the

$x$ and

$y$ values to find where the minimum occurs.

$(- \frac{3}{10}, - \frac{149}{20})$

Step 5

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