Algebra Examples

$- 9 x^{2} + 778 x - 7904$

Step 1

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{778}{2 (- 9)}$

Step 2.2

Remove parentheses.

$x = - \frac{778}{2 (- 9)}$

Step 2.3

Simplify $- \frac{778}{2 (- 9)}$ .

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Step 2.3.1

Cancel the common factor of $778$ and $2$ .

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Step 2.3.1.1

Factor $2$ out of $778$ .

$x = - \frac{2 \cdot 389}{2 \cdot - 9}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

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

$x = - \frac{2 \cdot 389}{2 (- 9)}$

Step 2.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot 389}{2 \cdot - 9}$

Step 2.3.1.2.3

Rewrite the expression.

$x = - \frac{389}{- 9}$

Step 2.3.2

Move the negative in front of the fraction.

$x = - - \frac{389}{9}$

Step 2.3.3

Multiply $- - \frac{389}{9}$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{389}{9})$

Step 2.3.3.2

Multiply $\frac{389}{9}$ by $1$ .

$x = \frac{389}{9}$

Step 3

Evaluate $f (\frac{389}{9})$ .

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Step 3.1

Replace the variable $x$ with $\frac{389}{9}$ in the expression.

$f (\frac{389}{9}) = - 9 {(\frac{389}{9})}^{2} + 778 (\frac{389}{9}) - 7904$

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 $\frac{389}{9}$ .

$f (\frac{389}{9}) = - 9 \frac{389^{2}}{9^{2}} + 778 (\frac{389}{9}) - 7904$

Step 3.2.1.2

Raise $389$ to the power of $2$ .

$f (\frac{389}{9}) = - 9 \frac{151321}{9^{2}} + 778 (\frac{389}{9}) - 7904$

Step 3.2.1.3

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

$f (\frac{389}{9}) = - 9 (\frac{151321}{81}) + 778 (\frac{389}{9}) - 7904$

Step 3.2.1.4

Cancel the common factor of $9$ .

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Step 3.2.1.4.1

Factor $9$ out of $- 9$ .

$f (\frac{389}{9}) = 9 (- 1) (\frac{151321}{81}) + 778 (\frac{389}{9}) - 7904$

Step 3.2.1.4.2

Factor $9$ out of $81$ .

$f (\frac{389}{9}) = 9 \cdot (- 1 \frac{151321}{9 \cdot 9}) + 778 (\frac{389}{9}) - 7904$

Step 3.2.1.4.3

Cancel the common factor.

$f (\frac{389}{9}) = 9 \cdot (- 1 \frac{151321}{9 \cdot 9}) + 778 (\frac{389}{9}) - 7904$

Step 3.2.1.4.4

Rewrite the expression.

$f (\frac{389}{9}) = - 1 (\frac{151321}{9}) + 778 (\frac{389}{9}) - 7904$

Step 3.2.1.5

Rewrite $- 1 (\frac{151321}{9})$ as $- (\frac{151321}{9})$ .

$f (\frac{389}{9}) = - (\frac{151321}{9}) + 778 (\frac{389}{9}) - 7904$

Step 3.2.1.6

Multiply $778 (\frac{389}{9})$ .

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Step 3.2.1.6.1

Combine $778$ and $\frac{389}{9}$ .

$f (\frac{389}{9}) = - \frac{151321}{9} + \frac{778 \cdot 389}{9} - 7904$

Step 3.2.1.6.2

Multiply $778$ by $389$ .

$f (\frac{389}{9}) = - \frac{151321}{9} + \frac{302642}{9} - 7904$

Step 3.2.2

Combine fractions.

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Step 3.2.2.1

Combine the numerators over the common denominator.

$f (\frac{389}{9}) = - 7904 + \frac{- 151321 + 302642}{9}$

Step 3.2.2.2

Add $- 151321$ and $302642$ .

$f (\frac{389}{9}) = - 7904 + \frac{151321}{9}$

Step 3.2.3

To write $- 7904$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{389}{9}) = - 7904 \cdot \frac{9}{9} + \frac{151321}{9}$

Step 3.2.4

Combine $- 7904$ and $\frac{9}{9}$ .

$f (\frac{389}{9}) = \frac{- 7904 \cdot 9}{9} + \frac{151321}{9}$

Step 3.2.5

Combine the numerators over the common denominator.

$f (\frac{389}{9}) = \frac{- 7904 \cdot 9 + 151321}{9}$

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 7904$ by $9$ .

$f (\frac{389}{9}) = \frac{- 71136 + 151321}{9}$

Step 3.2.6.2

Add $- 71136$ and $151321$ .

$f (\frac{389}{9}) = \frac{80185}{9}$

Step 3.2.7

The final answer is $\frac{80185}{9}$ .

$\frac{80185}{9}$

Step 4

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

$(\frac{389}{9}, \frac{80185}{9})$

Step 5