In the xy-plane, the graphs of \(\mathrm{y = (x - 1)^2}\) and 6x + y + 3 = 0 intersect...

1. TRANSLATE the problem information

• Given information:
  • Parabola: \(\mathrm{y = (x - 1)^2}\)
  • Line: \(\mathrm{6x + y + 3 = 0}\)
  • These graphs intersect at exactly one point
• We need to find the x-coordinate of this intersection point

2. INFER the solution approach

• When two graphs intersect, their y-values are equal at the intersection point
• Strategy: Solve for y from the linear equation, then set equal to the parabolic equation

3. SIMPLIFY the linear equation to isolate y

From \(\mathrm{6x + y + 3 = 0}\):

• \(\mathrm{y = -6x - 3}\)

4. INFER the key equation and SIMPLIFY

• Set the y-expressions equal: \(\mathrm{(x - 1)^2 = -6x - 3}\)
• Expand the left side: \(\mathrm{x^2 - 2x + 1 = -6x - 3}\)
• Move all terms to left side: \(\mathrm{x^2 - 2x + 1 + 6x + 3 = 0}\)
• Combine like terms: \(\mathrm{x^2 + 4x + 4 = 0}\)

5. INFER the factoring pattern and SIMPLIFY

• Recognize this as a perfect square: \(\mathrm{(x + 2)^2 = 0}\)
• Therefore: \(\mathrm{x + 2 = 0}\), so \(\mathrm{x = -2}\)

Answer: B. -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x - 1)^2}\) or when combining like terms, leading to an incorrect quadratic equation.

For example, they might incorrectly expand to get \(\mathrm{x^2 - 2x - 1}\) instead of \(\mathrm{x^2 - 2x + 1}\), or make sign errors when moving terms across the equation. These errors cascade through to produce wrong values when factoring and solving.

This may lead them to select Choice A (-6) or Choice D (3) depending on the specific algebraic mistake made.

Second Most Common Error:

Poor INFER reasoning about intersection: Students don't recognize that intersection means the y-values must be equal, so they attempt to solve the system using elimination or other methods instead of substitution.

This leads to confusion about how to proceed systematically, causing them to get stuck and guess among the answer choices.

The Bottom Line:

This problem tests whether students can connect the geometric concept of graph intersection to the algebraic process of solving equations, then execute the algebra cleanly without computational errors.