Question:x = 10\(\mathrm{x = -2(y - 4)^2 + 8}\)If the given equations are graphed in the xy-plane, at how many...

1. TRANSLATE the problem information

• Given equations:
  • \(\mathrm{x = 10}\) (vertical line)
  • \(\mathrm{x = -2(y - 4)^2 + 8}\) (parabola)
• Need to find: Number of intersection points

2. INFER the solution approach

• Since we want intersections, we need points that satisfy both equations simultaneously
• Both expressions equal x, so we can set them equal to each other: \(\mathrm{10 = -2(y - 4)^2 + 8}\)

3. SIMPLIFY the equation

• Start with: \(\mathrm{10 = -2(y - 4)^2 + 8}\)
• Subtract 8 from both sides: \(\mathrm{2 = -2(y - 4)^2}\)
• Divide both sides by -2: \(\mathrm{-1 = (y - 4)^2}\)

4. INFER the final conclusion

• We need \(\mathrm{(y - 4)^2 = -1}\)
• But squares of real numbers are always non-negative: \(\mathrm{(y - 4)^2 \geq 0}\)
• Since \(\mathrm{-1 \lt 0}\), this equation has no real solutions
• No solutions means no intersection points

Answer: D (Zero)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that squares of real numbers cannot be negative

Students correctly arrive at \(\mathrm{(y - 4)^2 = -1}\) but then attempt to solve it by taking the square root of both sides, writing \(\mathrm{y - 4 = \pm\sqrt{-1}}\). They might either get confused by the imaginary numbers or incorrectly think this gives real solutions. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Setting up the system incorrectly

Some students might try to solve this by substituting the first equation into the second equation incorrectly, such as writing \(\mathrm{10 = -2(10 - 4)^2 + 8}\), thinking they should substitute \(\mathrm{x = 10}\) into the y-variable position. This algebraic error leads them away from the systematic approach and typically results in guessing.

The Bottom Line:

This problem tests whether students understand both the systematic approach to finding intersections (setting equal expressions equal) and the fundamental property that squares of real numbers are non-negative. The key insight is recognizing when an equation has no real solutions.