To rewrite the quadratic equation [tex]\( y = -3x^2 - 12x - 2 \)[/tex] in vertex form, we need to complete the square. Let's work through this step-by-step process:
### Step 1: Rewrite the quadratic term and linear coefficient
Focus on the quadratic term and the linear coefficient:
[tex]\[ y = -3x^2 - 12x - 2 \][/tex]
Factor out the coefficient of [tex]\( x^2 \)[/tex] from these two terms:
[tex]\[ y = -3(x^2 + 4x) - 2 \][/tex]
### Step 2: Complete the square
To complete the square inside the parentheses, consider the expression [tex]\( x^2 + 4x \)[/tex]. We perform the following steps:
1. Take half of the linear coefficient (4), which is 2.
2. Square this value: [tex]\( 2^2 = 4 \)[/tex].
Add and subtract this square inside the parentheses:
[tex]\[ y = -3(x^2 + 4x + 4 - 4) - 2 \][/tex]
[tex]\[ y = -3((x^2 + 4x + 4) - 4) - 2 \][/tex]
### Step 3: Simplify the expression inside the parentheses
We can now write the completed square term as a perfect square:
[tex]\[ y = -3((x + 2)^2 - 4) - 2 \][/tex]
### Step 4: Distribute the -3
Distribute -3 to both terms inside the parentheses:
[tex]\[ y = -3(x + 2)^2 - 3(-4) - 2 \][/tex]
[tex]\[ y = -3(x + 2)^2 + 12 - 2 \][/tex]
### Step 5: Simplify
Combine the constants to simplify the equation:
[tex]\[ y = -3(x + 2)^2 + 10 \][/tex]
Thus, the vertex form of the equation is:
[tex]\[ y = -3(x + 2)^2 + 10 \][/tex]
From the given options:
1. [tex]\( y = -3(x + 2)^2 + 10 \)[/tex]
2. [tex]\( y = -3(x - 2)^2 + 10 \)[/tex]
3. [tex]\( y = -3(x + 2)^2 - 14 \)[/tex]
4. [tex]\( y = -3(x - 2)^2 - 2 \)[/tex]
The correct option is:
[tex]\[ y = -3(x + 2)^2 + 10 \][/tex]
So, the equation [tex]\( y = -3x^2 - 12x - 2 \)[/tex] rewritten in vertex form is:
[tex]\[ y = -3(x + 2)^2 + 10 \][/tex]
The correct answer is:
[tex]\[ 1 \][/tex]
