The Discriminant
The Discriminant with Parameters
A parameter is a letter inside a coefficient. Instead of producing one fixed discriminant value, substitution produces an expression in that parameter.
Parameters in quadratic coefficients
In x² + kx + 9 = 0, the coefficients are a = 1, b = k, and c = 9. The discriminant is therefore k² − 36.
Build Δ in terms of a parameter
- Identify a, b, and c, including the parameter.
- Substitute them into b² − 4ac.
- Simplify without treating the parameter as a known number.
- Use the resulting expression to study the roots.
Worked example
Find Δ for x² − 5x + m = 0
- a = 1, b = −5, c = m
- Δ = (−5)² − 4(1)(m)
Δ = 25 − 4m.
Common mistakes
- Treating the parameter as zero.
- Forgetting to square a parameter when it is b.
- Dropping the parameter during simplification.
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