Find the Vertex Form f(x)=6x^2+48x+4

$f (x) = 6 x^{2} + 48 x + 4$

Step 1

Write $f (x) = 6 x^{2} + 48 x + 4$ as an equation.

$y = 6 x^{2} + 48 x + 4$

Step 2

Complete the square for $6 x^{2} + 48 x + 4$ .

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

$a = 6$

$b = 48$

$c = 4$

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 $d = \frac{b}{2 a}$ .

$d = \frac{48}{2 \cdot 6}$

Simplify the right side.

Cancel the common factor of $48$ and $2$ .

Factor $2$ out of $48$ .

$d = \frac{2 \cdot 24}{2 \cdot 6}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 6$ .

$d = \frac{2 \cdot 24}{2 (6)}$

Cancel the common factor.

$d = \frac{2 \cdot 24}{2 \cdot 6}$

Rewrite the expression.

$d = \frac{24}{6}$

Cancel the common factor of $24$ and $6$ .

Factor $6$ out of $24$ .

$d = \frac{6 \cdot 4}{6}$

Cancel the common factors.

Factor $6$ out of $6$ .

$d = \frac{6 \cdot 4}{6 (1)}$

Cancel the common factor.

$d = \frac{6 \cdot 4}{6 \cdot 1}$

Rewrite the expression.

$d = \frac{4}{1}$

Divide $4$ by $1$ .

$d = 4$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 4 - \frac{48^{2}}{4 \cdot 6}$

Simplify the right side.

Simplify each term.

Raise $48$ to the power of $2$ .

$e = 4 - \frac{2304}{4 \cdot 6}$

Multiply $4$ by $6$ .

$e = 4 - \frac{2304}{24}$

Divide $2304$ by $24$ .

$e = 4 - 1 \cdot 96$

Multiply $- 1$ by $96$ .

$e = 4 - 96$

Subtract $96$ from $4$ .

$e = - 92$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $6 {(x + 4)}^{2} - 92$ .

$6 {(x + 4)}^{2} - 92$

Step 3

Set $y$ equal to the new right side.

$y = 6 {(x + 4)}^{2} - 92$

Find the Vertex Form f(x)=6x^2+48x+4

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