Find the Vertex Form g(x)=3x^2+24x+41

$g (x) = 3 x^{2} + 24 x + 41$

Step 1

Write $g (x) = 3 x^{2} + 24 x + 41$ as an equation.

$y = 3 x^{2} + 24 x + 41$

Step 2

Complete the square for $3 x^{2} + 24 x + 41$ .

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

$a = 3$

$b = 24$

$c = 41$

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

$d = \frac{24}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $24$ and $2$ .

Factor $2$ out of $24$ .

$d = \frac{2 \cdot 12}{2 \cdot 3}$

Cancel the common factors.

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

$d = \frac{2 \cdot 12}{2 (3)}$

Cancel the common factor.

$d = \frac{2 \cdot 12}{2 \cdot 3}$

Rewrite the expression.

$d = \frac{12}{3}$

Cancel the common factor of $12$ and $3$ .

Factor $3$ out of $12$ .

$d = \frac{3 \cdot 4}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$d = \frac{3 \cdot 4}{3 (1)}$

Cancel the common factor.

$d = \frac{3 \cdot 4}{3 \cdot 1}$

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

$e = 41 - \frac{24^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $24$ to the power of $2$ .

$e = 41 - \frac{576}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 41 - \frac{576}{12}$

Divide $576$ by $12$ .

$e = 41 - 1 \cdot 48$

Multiply $- 1$ by $48$ .

$e = 41 - 48$

Subtract $48$ from $41$ .

$e = - 7$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 4)}^{2} - 7$ .

$3 {(x + 4)}^{2} - 7$

Step 3

Set $y$ equal to the new right side.

$y = 3 {(x + 4)}^{2} - 7$

Find the Vertex Form g(x)=3x^2+24x+41

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