To determine what number can be added to the right side of the equation [tex]\(y = -2(x + 2)^2 - 1\)[/tex] in order to change it to a function with exactly one real root, we need to first analyze how the quadratic function behaves.
### Step 1: Understand the function
The given quadratic function is:
[tex]\[ y = -2(x + 2)^2 - 1 \][/tex]
This is in the form [tex]\( y = a(x - h)^2 + k \)[/tex] where:
- [tex]\( a = -2 \)[/tex] (a negative number, so the parabola opens downwards)
- The vertex [tex]\((h, k)\)[/tex] of the parabola is [tex]\((h, k)\ = (-2, -1)\)[/tex]
### Step 2: Determine the vertex
The vertex form of a quadratic function shows the maximum or minimum value of the function:
- Since the parabola opens downwards ( [tex]\(a < 0\)[/tex] ), the vertex represents the maximum point.
- Here, the vertex is at [tex]\((-2, -1)\)[/tex], meaning the highest point on the parabola is at [tex]\(y = -1\)[/tex].
### Step 3: Understand the requirement
To have the function [tex]\(\ y = -2(x + 2)^2 - 1 \)[/tex] have exactly one real root, the quadratic must touch the x-axis at one point. This means the maximum or minimum point should be on the x-axis.
### Step 4: Determine the necessary shift
Since the maximum value (vertex) is at [tex]\((-2, -1)\)[/tex] and it needs to touch the x-axis at [tex]\(y = 0\)[/tex], we need to raise the whole function by adding a constant [tex]\(C\)[/tex] so that the maximum point shifts to [tex]\(y = 0\)[/tex].
### Step 5: Calculation
Initially, the maximum point is [tex]\(y = -1\)[/tex]. To bring this maximum point to [tex]\(y = 0\)[/tex], we need to add 1 to the right side of the equation:
[tex]\[
y + C = -2(x + 2)^2 - 1
\][/tex]
We set [tex]\(C\)[/tex] so that the maximum value becomes zero:
[tex]\[
-1 + C = 0 \implies C = 1
\][/tex]
### Final Answer
Hence, the number that needs to be added to the right side of the equation to create a quadratic function with exactly one real root is [tex]\(\boxed{1}\)[/tex].
