To convert the quadratic function [tex]\( p(x) = 21 + 24x + 6x^2 \)[/tex] into vertex form [tex]\( p(x) = a(x - h)^2 + k \)[/tex], follow these steps:
### Step 1: Identify coefficients
The given quadratic function is in the form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = 6 \)[/tex]
- [tex]\( b = 24 \)[/tex]
- [tex]\( c = 21 \)[/tex]
### Step 2: Calculate the vertex [tex]\( (h, k) \)[/tex]
The vertex form of a quadratic function is [tex]\( p(x) = a(x - h)^2 + k \)[/tex], where [tex]\( h \)[/tex] is given by [tex]\( h = -\frac{b}{2a} \)[/tex] and [tex]\( k \)[/tex] is given by [tex]\( k = c - \frac{b^2}{4a} \)[/tex].
1. Calculate [tex]\( h \)[/tex]:
[tex]\[
h = -\frac{b}{2a} = -\frac{24}{2 \cdot 6} = -\frac{24}{12} = -2
\][/tex]
2. Calculate [tex]\( k \)[/tex]:
[tex]\[
k = c - \frac{b^2}{4a} = 21 - \frac{24^2}{4 \cdot 6} = 21 - \frac{576}{24} = 21 - 24 = -3
\][/tex]
### Step 3: Construct the vertex form
Substituting [tex]\( a \)[/tex], [tex]\( h \)[/tex], and [tex]\( k \)[/tex] into the vertex form, we get:
[tex]\[
p(x) = 6(x - (-2))^2 + (-3)
\][/tex]
Simplify the negative signs:
[tex]\[
p(x) = 6(x + 2)^2 - 3
\][/tex]
### Final Answer
The vertex form of the given quadratic function is:
[tex]\[
p(x) = 6(x + 2)^2 - 3
\][/tex]
