The equations y = x^2 and y = 3x + 180 are graphed in the xy-plane. What is a possible...

1. TRANSLATE the intersection problem into mathematical action

• Given information:
  • Graph 1: \(\mathrm{y = x^2}\)
  • Graph 2: \(\mathrm{y = 3x + 180}\)
  • Need: x-coordinate where graphs intersect
• TRANSLATE tells us: At intersection points, both equations have the same y-value, so we set them equal: \(\mathrm{x^2 = 3x + 180}\)

2. SIMPLIFY to standard quadratic form

• Rearrange equation: \(\mathrm{x^2 = 3x + 180}\)
• Move all terms to one side: \(\mathrm{x^2 - 3x - 180 = 0}\)
• Now we have a quadratic in standard form that we can factor

3. SIMPLIFY through factoring

• Need two numbers that multiply to -180 and add to -3
• Check factor pairs of 180: \(\mathrm{12 \times 15 = 180}\)
• Since we need sum of -3: try +12 and -15
• Check: \(\mathrm{12 + (-15) = -3}\) ✓ and \(\mathrm{12 \times (-15) = -180}\) ✓
• Factor: \(\mathrm{(x + 12)(x - 15) = 0}\)

4. APPLY CONSTRAINTS using zero product property

• If \(\mathrm{(x + 12)(x - 15) = 0}\), then \(\mathrm{x + 12 = 0}\) or \(\mathrm{x - 15 = 0}\)
• Solve: \(\mathrm{x = -12}\) or \(\mathrm{x = 15}\)
• Both values are valid intersection x-coordinates

Answer: B (-12) (Note: D (15) would also be a correct intersection point)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students struggle to find the correct factor pairs of -180 that sum to -3. They might try factors like 1 and 180, or 9 and 20, without systematically checking both the multiplication and addition requirements. This leads to incorrect factorization attempts like \(\mathrm{(x + 9)(x - 20)}\) or \(\mathrm{(x + 1)(x - 180)}\), which don't equal the original quadratic.

This causes them to get stuck and resort to guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might not recognize that intersection points occur where the equations are equal. Instead, they might try to solve each equation individually or attempt to graph without understanding that setting \(\mathrm{y = x^2}\) equal to \(\mathrm{y = 3x + 180}\) is the key step.

This leads to confusion and abandoning systematic solution approach.

The Bottom Line:

This problem tests whether students can connect the geometric concept of intersection with the algebraic technique of setting equations equal, then execute the factoring process systematically. Success requires both conceptual understanding and careful algebraic manipulation.