The Discriminant
Predicting Roots with the Discriminant
You can predict the number of real roots without fully solving the quadratic. Identify the coefficients, calculate Δ, and interpret its sign.
Why the discriminant predicts roots
The quadratic formula contains √Δ. A positive value has two square-root choices, zero has one value, and a negative value has no real square root.
Predict the number of roots
- Write the equation in standard form.
- Identify a, b, and c.
- Calculate Δ.
- Use the sign of Δ to state the number of real roots.
Worked example
Predict the roots of x² − 6x + 9 = 0
- a = 1, b = −6, c = 9
- Δ = (−6)² − 4(1)(9)
- Δ = 36 − 36 = 0
The equation has one repeated real root.
Common mistakes
- Solving for the roots when only their number is requested.
- Interpreting Δ = 0 as no roots.
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