Algebra Examples

$F (n) = 1.5 n^{2} - 4.5 n - 18$

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

$f_{min}$ $n = a n^{2} + b n + c$ occurs at $n = - \frac{b}{2 a}$

Step 2

Find the value of $n = - \frac{b}{2 a}$ .

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Step 2.1

Substitute in the values of $a$ and $b$ .

$n = - \frac{- 4.5}{2 (1.5)}$

Step 2.2

Remove parentheses.

$n = - \frac{- 4.5}{2 (1.5)}$

Step 2.3

Simplify $- \frac{- 4.5}{2 (1.5)}$ .

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Step 2.3.1

Multiply $2$ by $1.5$ .

$n = - \frac{- 4.5}{3}$

Step 2.3.2

Divide $- 4.5$ by $3$ .

$n = - - 1.5$

Step 2.3.3

Multiply $- 1$ by $- 1.5$ .

$n = 1.5$

Step 3

Evaluate $f (1.5)$ .

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Step 3.1

Replace the variable $n$ with $1.5$ in the expression.

$F (1.5) = 1.5 {(1.5)}^{2} - 4.5 \cdot 1.5 - 18$

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

Multiply $1.5$ by ${(1.5)}^{2}$ by adding the exponents.

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Step 3.2.1.1.1

Multiply $1.5$ by ${(1.5)}^{2}$ .

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Step 3.2.1.1.1.1

Raise $1.5$ to the power of $1$ .

$F (1.5) = 1.5 {(1.5)}^{2} - 4.5 \cdot 1.5 - 18$

Step 3.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$F (1.5) = {1.5}^{1 + 2} - 4.5 \cdot 1.5 - 18$

Step 3.2.1.1.2

Add $1$ and $2$ .

$F (1.5) = {1.5}^{3} - 4.5 \cdot 1.5 - 18$

Step 3.2.1.2

Raise $1.5$ to the power of $3$ .

$F (1.5) = 3.375 - 4.5 \cdot 1.5 - 18$

Step 3.2.1.3

Multiply $- 4.5$ by $1.5$ .

$F (1.5) = 3.375 - 6.75 - 18$

Step 3.2.2

Simplify by subtracting numbers.

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Step 3.2.2.1

Subtract $6.75$ from $3.375$ .

$F (1.5) = - 3.375 - 18$

Step 3.2.2.2

Subtract $18$ from $- 3.375$ .

$F (1.5) = - 21.375$

Step 3.2.3

The final answer is $- 21.375$ .

$- 21.375$

Step 4

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

$(1.5, - 21.375)$

Step 5