In the xy-plane, the graph of the equation y = x^2 + 3 is a parabola. A line with the...

1. TRANSLATE the tangency condition

• Given information:
  • Parabola: \(\mathrm{y = x^2 + 3}\)
  • Line: \(\mathrm{y = 2x + b}\)
  • The line is tangent to the parabola

• What this tells us: Tangent means the line intersects the parabola at exactly one point.

2. INFER the mathematical approach

• Key insight: When two curves intersect at exactly one point, setting their equations equal gives a quadratic with exactly one solution.
• Strategy: Set the equations equal, rearrange to standard form, then use the discriminant condition.

3. Set up the intersection equation

Set the parabola and line equations equal:

\(\mathrm{x^2 + 3 = 2x + b}\)

4. SIMPLIFY to standard quadratic form

Rearrange all terms to one side:

\(\mathrm{x^2 - 2x + (3 - b) = 0}\)

5. INFER the discriminant condition

• For exactly one solution, discriminant = 0
• In the form \(\mathrm{ax^2 + bx + c = 0}\), we have:
  • \(\mathrm{a = 1}\)
  • b coefficient = \(\mathrm{-2}\)
  • \(\mathrm{c = (3 - b)}\)

6. SIMPLIFY the discriminant equation

Apply \(\mathrm{\Delta = b^2 - 4ac = 0}\):

\(\mathrm{(-2)^2 - 4(1)(3 - b) = 0}\)

\(\mathrm{4 - 4(3 - b) = 0}\)

\(\mathrm{4 - 12 + 4b = 0}\)

\(\mathrm{-8 + 4b = 0}\)

\(\mathrm{4b = 8}\)

\(\mathrm{b = 2}\)

Answer: C. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect tangency to the "exactly one solution" condition. Instead, they might try to solve the system directly by substitution or attempt to find the actual intersection point coordinates first. This leads to unnecessary complexity and often abandoning the systematic approach, causing them to guess among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the discriminant equation but make algebraic errors, particularly when expanding \(\mathrm{-4(3 - b)}\) or combining like terms. A common mistake is getting the signs wrong: writing \(\mathrm{4 + 12 - 4b = 0}\) instead of \(\mathrm{4 - 12 + 4b = 0}\). This may lead them to select Choice A (-2) since they get \(\mathrm{b = -2}\).

The Bottom Line:

This problem tests whether students can translate a geometric concept (tangency) into an algebraic condition (discriminant = 0). The key breakthrough is recognizing that tangency doesn't require finding the actual point of tangency—just using the mathematical condition for exactly one intersection.