Sure, let's go through the steps to complete the square and write the given quadratic equation in vertex form.
Given equation:
[tex]\[ y = 3x^2 + 12x + 7 \][/tex]
We want to express it in the form [tex]\( y = a(x - h)^2 + k \)[/tex].
### Step-by-Step Solution:
1. Factor out the coefficient of [tex]\(x^2\)[/tex] from the first two terms:
[tex]\[ y = 3(x^2 + 4x) + 7 \][/tex]
2. Complete the square inside the parentheses:
- Take the coefficient of [tex]\(x\)[/tex] (which is 4), divide by 2, and square it. This gives [tex]\((4/2)^2 = 4\)[/tex].
- Add and subtract this value inside the parentheses:
[tex]\[ y = 3(x^2 + 4x + 4 - 4) + 7 \][/tex]
[tex]\[ y = 3((x^2 + 4x + 4) - 4) + 7 \][/tex]
3. Group the complete square and simplify:
[tex]\[ y = 3((x + 2)^2 - 4) + 7 \][/tex]
4. Distribute the 3:
[tex]\[ y = 3(x + 2)^2 - 12 + 7 \][/tex]
5. Simplify the constants:
[tex]\[ y = 3(x + 2)^2 - 5 \][/tex]
Now, the equation is in the vertex form [tex]\( y = a(x - h)^2 + k \)[/tex], where:
- [tex]\( a = 3 \)[/tex]
- [tex]\( h = -2 \)[/tex]
- [tex]\( k = -5 \)[/tex]
So, when the expression is written in vertex form:
- [tex]\( a \)[/tex] is [tex]\( 3 \)[/tex],
- [tex]\( h \)[/tex] is [tex]\( -2 \)[/tex],
- [tex]\( k \)[/tex] is [tex]\( -5 \)[/tex].
[tex]\[
\boxed{3} \quad \boxed{-2} \quad \boxed{-5} \checkmark
\][/tex]
