Find the Maximum/Minimum Value 3x^2-54x+241

$3 x^{2} - 54 x + 241$

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 $x = - \frac{b}{2 a}$ .

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

$x = - \frac{- 54}{2 (3)}$

Remove parentheses.

$x = - \frac{- 54}{2 (3)}$

Simplify $- \frac{- 54}{2 (3)}$ .

Cancel the common factor of $- 54$ and $2$ .

Factor $2$ out of $- 54$ .

$x = - \frac{2 \cdot - 27}{2 \cdot 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

$x = - \frac{2 \cdot - 27}{2 (3)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 27}{2 \cdot 3}$

Rewrite the expression.

$x = - \frac{- 27}{3}$

Cancel the common factor of $- 27$ and $3$ .

Factor $3$ out of $- 27$ .

$x = - \frac{3 \cdot - 9}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$x = - \frac{3 \cdot - 9}{3 (1)}$

Cancel the common factor.

$x = - \frac{3 \cdot - 9}{3 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 9}{1}$

Divide $- 9$ by $1$ .

$x = - - 9$

Multiply $- 1$ by $- 9$ .

$x = 9$

Step 3

Evaluate $f (9)$ .

Replace the variable $x$ with $9$ in the expression.

$f (9) = 3 {(9)}^{2} - 54 \cdot 9 + 241$

Simplify the result.

Simplify each term.

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

$f (9) = 3 \cdot 81 - 54 \cdot 9 + 241$

Multiply $3$ by $81$ .

$f (9) = 243 - 54 \cdot 9 + 241$

Multiply $- 54$ by $9$ .

$f (9) = 243 - 486 + 241$

Simplify by adding and subtracting.

Subtract $486$ from $243$ .

$f (9) = - 243 + 241$

Add $- 243$ and $241$ .

$f (9) = - 2$

The final answer is $- 2$ .

$- 2$

Step 4

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

$(9, - 2)$

Step 5

Find the Maximum/Minimum Value 3x^2-54x+241

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