Question:The equation \(\mathrm{y = (x - 6)^2 + 4}\) represents a parabola in the xy-plane.How many distinct x-intercepts does this...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{y = (x - 6)^2 + 4}\)
  • Need to find: number of distinct x-intercepts

• What this tells us: We need to find where this parabola crosses the x-axis

2. INFER the approach

• X-intercepts occur where the graph crosses the x-axis, meaning \(\mathrm{y = 0}\)
• We can solve this algebraically by setting \(\mathrm{y = 0}\), or analyze it using vertex form properties
• Let's use the algebraic method first

3. SIMPLIFY by setting up the equation

• Set \(\mathrm{y = 0}\):

\(\mathrm{0 = (x - 6)^2 + 4}\)

• Solve for \(\mathrm{(x - 6)^2}\):

\(\mathrm{(x - 6)^2 = -4}\)

4. INFER the meaning of the result

• We need \(\mathrm{(x - 6)^2 = -4}\)
• Since any real number squared gives a non-negative result, there's no real number x that satisfies this equation
• Therefore, there are zero real solutions

5. INFER verification using vertex form analysis

• The equation \(\mathrm{y = (x - 6)^2 + 4}\) is in vertex form \(\mathrm{y = a(x - h)^2 + k}\)
• Vertex is at \(\mathrm{(h, k) = (6, 4)}\)
• Since \(\mathrm{a = 1 \gt 0}\), the parabola opens upward
• The minimum y-value is 4 (at the vertex)
• Since the entire parabola lies above \(\mathrm{y = 0}\) (the x-axis), it never intersects the x-axis

Answer: A. Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may attempt to solve \(\mathrm{(x - 6)^2 = -4}\) by taking the square root of both sides without recognizing that negative numbers don't have real square roots.

They might write:

\(\mathrm{x - 6 = ±\sqrt{-4} = ±2i}\)

then conclude there are two x-intercepts at \(\mathrm{x = 6 ± 2i}\). However, these are complex numbers, not real x-intercepts that can be graphed on a standard xy-plane.

This leads to confusion about whether complex solutions count as x-intercepts, potentially causing them to select Choice C (Exactly two).

Second Most Common Error:

Conceptual confusion about vertex form: Students might incorrectly think that because the equation can be written in vertex form, it automatically has x-intercepts. They may confuse the vertex coordinates with intercept information or misremember that "all parabolas have two x-intercepts."

This misconception may lead them to select Choice C (Exactly two) without actually solving the problem.

The Bottom Line:

This problem tests whether students understand that x-intercepts must be real numbers and can distinguish between the existence of complex solutions versus real x-intercepts on the coordinate plane. The key insight is recognizing when an equation has no real solutions.