To convert [tex]\( y = 6x^2 + 36x + 48 \)[/tex] into vertex form, we need to complete the square. Here's a detailed step-by-step solution:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the first two terms:
[tex]\[
y = 6(x^2 + 6x) + 48
\][/tex]
2. Complete the square inside the parentheses:
- Take half of the coefficient of [tex]\( x \)[/tex] (which is 6), square it, and add and subtract it inside the parentheses.
[tex]\[
\left(\frac{6}{2}\right)^2 = 9
\][/tex]
- Add and subtract 9 inside the parentheses:
[tex]\[
y = 6(x^2 + 6x + 9 - 9) + 48
\][/tex]
- Rewrite the expression inside the parentheses by completing the square:
[tex]\[
y = 6((x + 3)^2 - 9) + 48
\][/tex]
3. Simplify the equation by distributing the 6 and combining like terms:
- Distribute the 6:
[tex]\[
y = 6(x + 3)^2 - 6 \times 9 + 48
\][/tex]
- Simplify the constant term:
[tex]\[
y = 6(x + 3)^2 - 54 + 48
\][/tex]
- Combine like terms:
[tex]\[
y = 6(x + 3)^2 - 6
\][/tex]
Therefore, the equation in vertex form is:
[tex]\[
y = 6(x + 3)^2 - 6
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{b}
\][/tex]
