The equations y = 12 and y = x^2 + 5x + 16 are graphed in the xy-plane. The graphs...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = 12}\) (horizontal line)
  • Second equation: \(\mathrm{y = x^2 + 5x + 16}\) (parabola)
  • The graphs intersect at point (p, q)
  • Need to find a possible value of p (the x-coordinate)

• What this tells us: At intersection points, both equations must be satisfied by the same (x, y) values.

2. INFER the solution approach

• Key insight: At intersection points, the y-values from both equations are equal
• Strategy: Set the right sides of both equations equal to each other
• This will give us an equation in terms of x only, which we can solve for p

3. TRANSLATE and set up the equation

• Since \(\mathrm{y = 12}\) from the first equation and \(\mathrm{y = x^2 + 5x + 16}\) from the second:
  \(\mathrm{12 = x^2 + 5x + 16}\)

4. SIMPLIFY to standard quadratic form

• Subtract 16 from both sides:
  \(\mathrm{12 - 16 = x^2 + 5x}\)
  \(\mathrm{-4 = x^2 + 5x}\)

• Rearrange to standard form:
  \(\mathrm{x^2 + 5x + 4 = 0}\)

5. SIMPLIFY by factoring the quadratic

• Look for two numbers that multiply to 4 and add to 5
• Those numbers are 4 and 1
• Factor: \(\mathrm{(x + 4)(x + 1) = 0}\)

6. APPLY the zero product property and CONSIDER ALL CASES

• If \(\mathrm{(x + 4)(x + 1) = 0}\), then either factor equals zero:
  • \(\mathrm{x + 4 = 0}\), so \(\mathrm{x = -4}\)
  • \(\mathrm{x + 1 = 0}\), so \(\mathrm{x = -1}\)

• Both values are possible x-coordinates for intersection points

7. APPLY CONSTRAINTS by checking answer choices

• Looking at the options: (A) -6, (B) -4, (C) 3, (D) 12
• Our solutions \(\mathrm{x = -4}\) and \(\mathrm{x = -1}\) show that -4 appears as choice (B)

Answer: B (-4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't recognize that "graphs intersect" means setting the equations equal to each other. Instead, they might try to graph both equations or substitute random values.

Without this key translation, they get stuck trying to work with two separate equations and often abandon systematic solution, leading to guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{12 = x^2 + 5x + 16}\) but make algebraic errors when rearranging. Common mistakes include:

• Incorrectly getting \(\mathrm{x^2 + 5x - 4 = 0}\) instead of \(\mathrm{x^2 + 5x + 4 = 0}\)
• Factoring errors, such as \(\mathrm{(x + 2)(x + 2) = 0}\)

These algebraic mistakes lead to wrong x-values that don't match any answer choice, causing confusion and guessing.

The Bottom Line:

This problem tests whether students understand the fundamental concept that intersection points satisfy both equations simultaneously, requiring them to set equations equal. The algebraic manipulation is straightforward once this key insight is made.