y = 18 \(\mathrm{y = -3(x - 18)^2 + 15}\) If the given equations are graphed in the xy-plane, at...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = 18}\) (horizontal line)
  • Second equation: \(\mathrm{y = -3(x - 18)^2 + 15}\) (parabola opening downward)
  • Need to find: Number of intersection points
• What this tells us: We need to solve this system of equations to find where the graphs meet.

2. INFER the solution approach

• Since we have \(\mathrm{y = 18}\) from the first equation, we can substitute this value directly into the second equation
• This will give us an equation in terms of x only, which we can solve

3. SIMPLIFY through substitution and algebraic manipulation

• Substitute \(\mathrm{y = 18}\) into the second equation:
  \(\mathrm{18 = -3(x - 18)^2 + 15}\)

• Subtract 15 from both sides:
  \(\mathrm{18 - 15 = -3(x - 18)^2}\)
  \(\mathrm{3 = -3(x - 18)^2}\)

• Divide both sides by -3:
  \(\mathrm{3 ÷ (-3) = (x - 18)^2}\)
  \(\mathrm{-1 = (x - 18)^2}\)

4. INFER the mathematical impossibility

• We now have \(\mathrm{(x - 18)^2 = -1}\)
• Since \(\mathrm{(x - 18)}\) represents some real number, and the square of any real number is always non-negative (\(\mathrm{≥ 0}\)), this equation has no real solutions
• No real solutions means the graphs never intersect

Answer: D. Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that \(\mathrm{(x - 18)^2 = -1}\) is impossible for real numbers. They might try to solve it by taking the square root of both sides, writing \(\mathrm{x - 18 = ±\sqrt{-1}}\), and then either get confused about what to do with \(\mathrm{\sqrt{-1}}\) or assume there are two solutions.

This leads to confusion and guessing, often selecting Choice B (Exactly two) because they think "plus or minus" means two solutions.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make algebraic errors during the manipulation steps. For example, they might incorrectly handle the signs when dividing by -3, getting \(\mathrm{+1 = (x - 18)^2}\) instead of \(\mathrm{-1 = (x - 18)^2}\). This would give them real solutions \(\mathrm{x = 17}\) or \(\mathrm{x = 19}\), leading them to believe there are intersection points.

This may lead them to select Choice B (Exactly two).

The Bottom Line:

This problem tests whether students can recognize when a mathematical equation has no real solutions. The key insight is understanding that squares of real numbers cannot be negative - a fundamental property that determines the entire outcome.