Math Problem Statement

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?

Solution

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

  • Line: x+2y+2=0x + 2y + 2 = 0
  • Parabola: y=(x1)2y = (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=0x + 2y + 2 = 0, solve for yy: y=x+22y = -\frac{x + 2}{2}

  2. Substitute this into the parabola equation:
    Set y=(x1)2y = (x - 1)^2, so: x+22=(x1)2-\frac{x + 2}{2} = (x - 1)^2 Multiply both sides by 2 to eliminate the fraction: (x+2)=2(x1)2-(x + 2) = 2(x - 1)^2 Simplify: (x+2)=2(x22x+1)-(x + 2) = 2(x^2 - 2x + 1) Expand: (x+2)=2x24x+2-(x + 2) = 2x^2 - 4x + 2 Rearrange to form a quadratic equation: 0=2x24x+2+x+20 = 2x^2 - 4x + 2 + x + 2 0=2x23x+40 = 2x^2 - 3x + 4

  3. Solve the quadratic equation:
    We now solve 2x23x+4=02x^2 - 3x + 4 = 0. The discriminant Δ\Delta of this quadratic equation is: Δ=(3)24(2)(4)=932=23\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

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Intersection of Curves

Formulas

Quadratic equation ax^2 + bx + c = 0
Discriminant formula Δ = b^2 - 4ac

Theorems

Quadratic formula
Discriminant Analysis for Intersection

Suitable Grade Level

Grades 10-12