Parameter Problems
Finding Values of k for Real Roots
Questions about values of k are often discriminant problems. Build Δ in terms of k, then apply the condition for the requested number of roots.
Connect k to the number of roots
The parameter changes a coefficient, which changes the discriminant. Solving a condition on Δ tells you exactly which k-values create each root case.
Find the allowed values of k
- Identify a, b, and c in terms of k.
- Simplify Δ = b² − 4ac.
- Set Δ > 0, Δ = 0, or Δ < 0.
- Solve for k and state the complete set of values.
Worked example
Two roots for x² + kx + 9 = 0
- Δ = k² − 36
- Two distinct roots require k² − 36 > 0.
- So k² > 36.
There are two distinct real roots when k < −6 or k > 6.
Common mistakes
- Giving only k > 6 and missing the negative interval.
- Using Δ ≥ 0 when roots must be distinct.
- Confusing k-values with the equation’s x-roots.
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