Worked example
x² + 4x + 5 = 0
- Identify a = 1, b = 4, c = 5.
- Δ = 16 − 20 = −4.
- Because Δ is negative, there are no real roots.
- In the complex numbers, the roots are −2 ± i.
Math tool
The discriminant predicts the real-root pattern.
Calculate b² − 4ac and understand what its sign says about the real roots of a quadratic.
For real coefficients, Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives no real roots. It does not replace solving when the actual roots are needed.
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