Question
Use the graphs to fill in the blanks.
x^2-4x+4
x^2+6x+9
1
2
3
4
5
-10
2
X
10
X
-10
-3
-20
Zero:
ยก Answer
-;
Zero:
Answer
Repeated Factor:
1 Answer
Repeated Factor:
Answer :
-3
(x+3)
; (x- 2)
::2
x^2-4x+4
x^2+6x+9
1
2
3
4
5
-10
2
X
10
X
-10
-3
-20
Zero:
ยก Answer
-;
Zero:
Answer
Repeated Factor:
1 Answer
Repeated Factor:
Answer :
-3
(x+3)
; (x- 2)
::2
Answers
GPT-4o mini
To analyze the quadratic expressions and fill in the blanks provided, let's break down the two quadratic expressions:
1. **For \(x^2 - 4x + 4\)**:
- This can be factored as:
\[
(x - 2)^2
\]
- This means it has a repeated factor of \(x - 2\).
- The zero (or root) of this equation can be found by setting it to zero:
\[
x - 2 = 0 \implies x = 2
\]
2. **For \(x^2 + 6x + 9\)**:
- This can be factored as:
\[
(x + 3)^2
\]
- This means it also has a repeated factor of \(x + 3\).
- The zero (or root) of this equation can be found by setting it to zero:
\[
x + 3 = 0 \implies x = -3
\]
Now we can summarize the findings for the blanks:
- **Zero for \(x^2 - 4x + 4\)**: \(2\)
- **Zero for \(x^2 + 6x + 9\)**: \(-3\)
- **Repeated Factor for \(x^2 - 4x + 4\)**: \(x - 2\)
- **Repeated Factor for \(x^2 + 6x + 9\)**: \(x + 3\)
Based on this, the answers can be filled in as follows:
- Zero: **2**
- Zero: **-3**
- Repeated Factor: **(x - 2)**
- Repeated Factor: **(x + 3)**
Please ensure robustness by cross-verifying these calculations and identifications against the graphs referred to, since I have made the conclusions purely from algebraic manipulations of the polynomials.
1. **For \(x^2 - 4x + 4\)**:
- This can be factored as:
\[
(x - 2)^2
\]
- This means it has a repeated factor of \(x - 2\).
- The zero (or root) of this equation can be found by setting it to zero:
\[
x - 2 = 0 \implies x = 2
\]
2. **For \(x^2 + 6x + 9\)**:
- This can be factored as:
\[
(x + 3)^2
\]
- This means it also has a repeated factor of \(x + 3\).
- The zero (or root) of this equation can be found by setting it to zero:
\[
x + 3 = 0 \implies x = -3
\]
Now we can summarize the findings for the blanks:
- **Zero for \(x^2 - 4x + 4\)**: \(2\)
- **Zero for \(x^2 + 6x + 9\)**: \(-3\)
- **Repeated Factor for \(x^2 - 4x + 4\)**: \(x - 2\)
- **Repeated Factor for \(x^2 + 6x + 9\)**: \(x + 3\)
Based on this, the answers can be filled in as follows:
- Zero: **2**
- Zero: **-3**
- Repeated Factor: **(x - 2)**
- Repeated Factor: **(x + 3)**
Please ensure robustness by cross-verifying these calculations and identifications against the graphs referred to, since I have made the conclusions purely from algebraic manipulations of the polynomials.