IB DP1 HL Mathematics
Practice Worksheet – Quadratic Equations with
Parameters
Instructions:
For each question, discuss the number of real roots of the given quadratic. Use the discriminant
∆ = b² - 4ac. Clearly state the conditions for:
- Two distinct real roots (∆ > 0)
- One repeated real root (∆ = 0)
- No real roots (∆ < 0)
Part A: Standard Quadratics
1. x² - 5x + 6 = 0
2. 2x² + 4x + 7 = 0
3. 3x² - 12x + 12 = 0
4. x² - 2kx + (k² - 1) = 0
Part B: Quadratics with Parameters
For each, find the condition(s) on the parameter for:
- Two distinct real roots
- One repeated real root
- No real roots
1. x² + (m-2)x + m = 0
2. (p+1)x² - 4x + (p-3) = 0
3. kx² + 2(k-1)x + 1 = 0, where k ≠ 0
4. (a²-1)x² + 2ax + 3 = 0
Part C: Application Problems
1. A quadratic function is defined as f(x) = x² - (k+3)x + (2k+1). Find the values of k for which f(x)
has exactly one real root.
2. A parabola is given by y = tx² - 2x + 1.
(a) Find the condition on t so that the parabola cuts the x-axis at two distinct points.
(b) Find the condition on t so that the parabola just touches the x-axis.
3. A quadratic equation x² + px + q = 0 has two distinct real roots. Show that p² > 4q.
Part D: Extension Challenge
Consider the quadratic (k+1)x² - 2kx + (k-3) = 0.
1. For what values of k does the quadratic have two distinct real roots?
2. For what values of k does the quadratic have exactly one solution?
3. For what values of k does the quadratic have no real roots?
✍■ Show all working clearly, and state your final conditions in terms of the parameter(s).
