Find the Vertex 3x^2+54x+240

$3 x^{2} + 54 x + 240$

Step 1

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} + 54 x + 240$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 3$

$b = 54$

$c = 240$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{54}{2 \cdot 3}$

Simplify the right side.

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

Factor $2$ out of $54$ .

$d = \frac{2 \cdot 27}{2 \cdot 3}$

Cancel the common factors.

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

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

Cancel the common factor.

$d = \frac{2 \cdot 27}{2 \cdot 3}$

Rewrite the expression.

$d = \frac{27}{3}$

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

Factor $3$ out of $27$ .

$d = \frac{3 \cdot 9}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

$d = \frac{3 \cdot 9}{3 \cdot 1}$

Rewrite the expression.

$d = \frac{9}{1}$

Divide $9$ by $1$ .

$d = 9$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 240 - \frac{54^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $54$ to the power of $2$ .

$e = 240 - \frac{2916}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 240 - \frac{2916}{12}$

Divide $2916$ by $12$ .

$e = 240 - 1 \cdot 243$

Multiply $- 1$ by $243$ .

$e = 240 - 243$

Subtract $243$ from $240$ .

$e = - 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 9)}^{2} - 3$ .

$3 {(x + 9)}^{2} - 3$

Set $y$ equal to the new right side.

$y = 3 {(x + 9)}^{2} - 3$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 3$

$h = - 9$

$k = - 3$

Step 3

Find the vertex $(h, k)$ .

$(- 9, - 3)$

Step 4

Find the Vertex 3x^2+54x+240

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