https://file.aiquickdraw.com/mathbot/file/17289842218623c00sddu.jpg

The image presents a mathematical problem involving the intersection of a straight line and a parabola. The given equations are:

- Line: \( x + 2y + 2 = 0 \)
- Parabola: \( y = (x - 1)^2 \)

The question asks how many intersection points these two equations have, and provides multiple-choice options: 
A. Two distinct intersection points  
B. Four distinct intersection points  
C. No intersection point  
D. A unique intersection point  
E. Three distinct intersection points

### Solution:

1. **Rearrange the line equation**:  
   From \( x + 2y + 2 = 0 \), solve for \( y \):
   \[
   y = -\frac{x + 2}{2}
   \]

2. **Substitute this into the parabola equation**:  
   Set \( y = (x - 1)^2 \), so:
   \[
   -\frac{x + 2}{2} = (x - 1)^2
   \]
   Multiply both sides by 2 to eliminate the fraction:
   \[
   -(x + 2) = 2(x - 1)^2
   \]
   Simplify:
   \[
   -(x + 2) = 2(x^2 - 2x + 1)
   \]
   Expand:
   \[
   -(x + 2) = 2x^2 - 4x + 2
   \]
   Rearrange to form a quadratic equation:
   \[
   0 = 2x^2 - 4x + 2 + x + 2
   \]
   \[
   0 = 2x^2 - 3x + 4
   \]

3. **Solve the quadratic equation**:  
   We now solve \( 2x^2 - 3x + 4 = 0 \). The discriminant \( \Delta \) of this quadratic equation is:
   \[
   \Delta = (-3)^2 - 4(2)(4) = 9 - 32 = -23
   \]
   Since the discriminant is negative, there are no real solutions.

Thus, the line and the parabola do not intersect at any point.

### Conclusion:
The correct answer is **C. No intersection point**.

Would you like further details or have any other questions?

Here are 5 related questions to explore:
1. How do you solve systems involving non-linear equations geometrically?
2. What does a negative discriminant mean in terms of intersection points?
3. How can you find the vertex of a parabola from its equation?
4. How do different types of conic sections intersect with lines?
5. What methods can be used to solve quadratic equations?

**Tip:** Always check the discriminant when solving quadratic equations to determine if real solutions exist

In the Cartesian plane, the straight line with equation x + 2y + 2 = 0 and the parabola with equation y = (x - 1)^2 have how many intersection points?

Find the number of intersection points between the line x + 2y + 2 = 0 and the parabola y = (x - 1)^2. Learn to solve the problem using quadratic equations and discriminant analysis. Suitable for high school students, Grades 10-12.

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