The Discriminant
Working Backward from the Number of Roots
The discriminant rule works in both directions. If you know how many real roots an equation has, you can state what must be true about Δ.
Reverse the discriminant rule
Two distinct real roots require Δ > 0, one repeated real root requires Δ = 0, and no real roots require Δ < 0.
Translate roots into a condition
- Identify the stated number of real roots.
- Match it to the sign condition for Δ.
- If a parameter is present, apply that condition to its discriminant expression.
Worked example
An equation has no real roots
- No real roots correspond to a negative discriminant.
The required condition is Δ < 0.
Common mistakes
- Reversing the rules for zero and negative discriminants.
- Writing a numerical discriminant when only its sign is known.
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