In the xy-plane, a line with equation 2y = 4.5 intersects a parabola at exactly one point. If the parabola...

1. TRANSLATE the problem information

• Given information:
  • Line equation: \(2\mathrm{y} = 4.5\)
  • Parabola equation: \(\mathrm{y} = -4\mathrm{x}^2 + \mathrm{bx}\)
  • They intersect at exactly one point
  • b is a positive constant

2. INFER what "exactly one point" means

• When a line intersects a parabola at exactly one point, they are tangent
• This means the system of equations has exactly one solution
• For a quadratic equation, this happens when the discriminant equals zero

3. SIMPLIFY the line equation

• Start with: \(2\mathrm{y} = 4.5\)
• Divide both sides by 2: \(\mathrm{y} = 2.25\)

4. SIMPLIFY to find the intersection condition

• Set the equations equal: \(2.25 = -4\mathrm{x}^2 + \mathrm{bx}\)
• Rearrange to standard form: \(4\mathrm{x}^2 - \mathrm{bx} + 2.25 = 0\)
• This gives us: \(\mathrm{a} = 4\), coefficient of x = \(-\mathrm{b}\), \(\mathrm{c} = 2.25\)

5. INFER and apply the discriminant condition

• For exactly one solution: discriminant = 0
• Discriminant formula: \(\mathrm{b}^2 - 4\mathrm{ac}\)
• Substitute: \((-\mathrm{b})^2 - 4(4)(2.25) = 0\)
• SIMPLIFY: \(\mathrm{b}^2 - 36 = 0\)

6. SIMPLIFY to solve for b

• \(\mathrm{b}^2 = 36\)
• \(\mathrm{b} = ±6\)

7. APPLY CONSTRAINTS to select final answer

• Since b is positive: \(\mathrm{b} = 6\)

Answer: 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "intersects at exactly one point" with the discriminant condition. They might try to solve the system directly by substitution without recognizing that the key insight is setting discriminant = 0. This leads to confusion about how to proceed systematically, causing them to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when setting up the quadratic or applying the discriminant formula. Common mistakes include sign errors when rearranging \(2.25 = -4\mathrm{x}^2 + \mathrm{bx}\), or incorrectly identifying the coefficients in the discriminant formula. This may lead them to get a wrong value for \(\mathrm{b}^2\) or make computational errors.

The Bottom Line:

The key insight is recognizing that geometric language ("intersects at exactly one point") translates to an algebraic condition (discriminant = 0). Without this connection, students resort to less efficient approaches or get overwhelmed by the system of equations.