Examples

$f (x) = 2 x^{2} + 3 x - 4$

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 $b$ .

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

Step 2.2

Remove parentheses.

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

Step 2.3

Multiply $2$ by $2$ .

$x = - \frac{3}{4}$

Step 3

Evaluate $f (- \frac{3}{4})$ .

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Step 3.1

Replace the variable $x$ with $- \frac{3}{4}$ in the expression.

$f (- \frac{3}{4}) = 2 {(- \frac{3}{4})}^{2} + 3 (- \frac{3}{4}) - 4$

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 $- \frac{3}{4}$ .

$f (- \frac{3}{4}) = 2 ({(- 1)}^{2} {(\frac{3}{4})}^{2}) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.1.2

Apply the product rule to $\frac{3}{4}$ .

$f (- \frac{3}{4}) = 2 ({(- 1)}^{2} (\frac{3^{2}}{4^{2}})) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

$f (- \frac{3}{4}) = 2 (1 (\frac{3^{2}}{4^{2}})) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.3

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$f (- \frac{3}{4}) = 2 (\frac{3^{2}}{4^{2}}) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.4

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

$f (- \frac{3}{4}) = 2 (\frac{9}{4^{2}}) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.5

Raise $4$ to the power of $2$ .

$f (- \frac{3}{4}) = 2 (\frac{9}{16}) + 3 (- \frac{3}{4}) - 4$

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 $16$ .

$f (- \frac{3}{4}) = 2 (\frac{9}{2 (8)}) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.6.2

Cancel the common factor.

$f (- \frac{3}{4}) = 2 (\frac{9}{2 \cdot 8}) + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.6.3

Rewrite the expression.

$f (- \frac{3}{4}) = \frac{9}{8} + 3 (- \frac{3}{4}) - 4$

Step 3.2.1.7

Multiply $3 (- \frac{3}{4})$ .

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Step 3.2.1.7.1

Multiply $- 1$ by $3$ .

$f (- \frac{3}{4}) = \frac{9}{8} - 3 (\frac{3}{4}) - 4$

Step 3.2.1.7.2

Combine $- 3$ and $\frac{3}{4}$ .

$f (- \frac{3}{4}) = \frac{9}{8} + \frac{- 3 \cdot 3}{4} - 4$

Step 3.2.1.7.3

Multiply $- 3$ by $3$ .

$f (- \frac{3}{4}) = \frac{9}{8} + \frac{- 9}{4} - 4$

Step 3.2.1.8

Move the negative in front of the fraction.

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9}{4} - 4$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Multiply $\frac{9}{4}$ by $\frac{2}{2}$ .

$f (- \frac{3}{4}) = \frac{9}{8} - (\frac{9}{4} \cdot \frac{2}{2}) - 4$

Step 3.2.2.2

Multiply $\frac{9}{4}$ by $\frac{2}{2}$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{4 \cdot 2} - 4$

Step 3.2.2.3

Write $- 4$ as a fraction with denominator $1$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{4 \cdot 2} + \frac{- 4}{1}$

Step 3.2.2.4

Multiply $\frac{- 4}{1}$ by $\frac{8}{8}$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{4 \cdot 2} + \frac{- 4}{1} \cdot \frac{8}{8}$

Step 3.2.2.5

Multiply $\frac{- 4}{1}$ by $\frac{8}{8}$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{4 \cdot 2} + \frac{- 4 \cdot 8}{8}$

Step 3.2.2.6

Reorder the factors of $4 \cdot 2$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{2 \cdot 4} + \frac{- 4 \cdot 8}{8}$

Step 3.2.2.7

Multiply $2$ by $4$ .

$f (- \frac{3}{4}) = \frac{9}{8} - \frac{9 \cdot 2}{8} + \frac{- 4 \cdot 8}{8}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- \frac{3}{4}) = \frac{9 - 9 \cdot 2 - 4 \cdot 8}{8}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $- 9$ by $2$ .

$f (- \frac{3}{4}) = \frac{9 - 18 - 4 \cdot 8}{8}$

Step 3.2.4.2

Multiply $- 4$ by $8$ .

$f (- \frac{3}{4}) = \frac{9 - 18 - 32}{8}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract $18$ from $9$ .

$f (- \frac{3}{4}) = \frac{- 9 - 32}{8}$

Step 3.2.5.2

Subtract $32$ from $- 9$ .

$f (- \frac{3}{4}) = \frac{- 41}{8}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (- \frac{3}{4}) = - \frac{41}{8}$

Step 3.2.6

The final answer is $- \frac{41}{8}$ .

$- \frac{41}{8}$

Step 4

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

$(- \frac{3}{4}, - \frac{41}{8})$

Step 5

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