The Discriminant
Discriminant Parameter Inequalities
Parameter problems combine the discriminant rule with an equation or inequality. The requested number of roots determines whether Δ must be positive, zero, or negative.
Turn a root condition into an inequality
For two distinct real roots, set Δ > 0. For one repeated root, set Δ = 0. For no real roots, set Δ < 0.
Solve a discriminant parameter problem
- Write Δ as an expression in the parameter.
- Choose > 0, = 0, or < 0 from the root condition.
- Solve the resulting equation or inequality.
- Check endpoints and interval notation.
Worked example
When does x² + kx + 9 = 0 have no real roots?
- Δ = k² − 36
- No real roots require k² − 36 < 0.
- So k² < 36.
The equation has no real roots when −6 < k < 6.
Common mistakes
- Using ≥ when the question requires distinct roots.
- Including endpoints when Δ must be strictly negative.
- Solving k² < 36 as only k < 6.
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