The equations y = x^2 + 3x - 5 and y = x^2 - 2x + 10 are graphed in...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = x^2 + 3x - 5}\)
  • Second equation: \(\mathrm{y = x^2 - 2x + 10}\)
  • Need to find: Number of intersection points

• What this tells us: We need to find where both equations give the same y-value for the same x-value

2. TRANSLATE the mathematical approach

• Key insight: Graphs intersect where they share the same coordinates
• This means: Set the right sides of both equations equal to each other
• Set up: \(\mathrm{x^2 + 3x - 5 = x^2 - 2x + 10}\)

3. SIMPLIFY by eliminating like terms

• Subtract \(\mathrm{x^2}\) from both sides:
  \(\mathrm{x^2 + 3x - 5 = x^2 - 2x + 10}\)
  \(\mathrm{3x - 5 = -2x + 10}\)

4. SIMPLIFY to solve the linear equation

• Add \(\mathrm{2x}\) to both sides:
  \(\mathrm{3x - 5 + 2x = -2x + 10 + 2x}\)
  \(\mathrm{5x - 5 = 10}\)

• Add \(\mathrm{5}\) to both sides:
  \(\mathrm{5x - 5 + 5 = 10 + 5}\)
  \(\mathrm{5x = 15}\)

• Divide by \(\mathrm{5}\):
  \(\mathrm{x = 3}\)

5. INFER the final answer

• Since we found exactly one x-value (\(\mathrm{x = 3}\)), there is exactly one point where the graphs intersect
• We don't need to find the y-coordinate to answer the question - we only need to count intersection points

Answer: B. 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "graphs intersect" means "set the equations equal to each other."

Instead, they might try to solve each equation separately, set each equal to zero, or attempt to graph both equations. This approach doesn't lead to finding intersection points and causes them to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{x^2 + 3x - 5 = x^2 - 2x + 10}\) but make algebraic errors when combining like terms.

Common mistake: When subtracting \(\mathrm{x^2}\) from both sides, they might incorrectly combine the linear terms, getting something like \(\mathrm{5x - 5 = 10}\) directly (skipping the \(\mathrm{-2x + 10}\) step). This could lead to a wrong x-value and confusion about the final answer, causing them to guess among the choices.

The Bottom Line:

This problem tests whether students can connect the geometric concept of "intersection" with the algebraic technique of "setting equations equal." The key breakthrough is realizing that you don't need to graph anything - algebra gives you the answer directly.