Two distinct real roots
x² − 3x − 4 = 0 has a discriminant of 25, giving roots x₁ = 4 and x₂ = −1.
Quadratic formula calculator
Result
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How it works
The discriminant, b² − 4ac, decides everything before you even reach the square root: positive means two distinct real roots, zero means one repeated real root (the parabola's vertex sits exactly on the x-axis), and negative means the roots are complex numbers, since a negative number has no real square root.
The ± in the formula is what produces two roots from one equation — one root uses +√Δ, the other −√Δ.
Formulas used
Quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
- a
- coefficient of x²
- b
- coefficient of x
- c
- constant term
Worked examples
No real roots
x² + 2x + 5 = 0 has a discriminant of −16, so its two roots are complex: −1 + 2i and −1 − 2i.
Assumptions and limits
- The equation is in standard form ax² + bx + c = 0.
- a, b and c are real numbers.
Frequently asked questions
What does a negative discriminant mean graphically?
The parabola described by the equation never touches the x-axis — it stays entirely above or entirely below it, so there is no real value of x where y = 0.
Can a be a fraction or negative?
Yes — the formula works for any nonzero value of a, positive, negative or a fraction. Only a = 0 breaks it, since the equation stops being quadratic.
What are complex roots used for?
They show up throughout engineering and physics, especially in problems involving oscillation or waves — even when a real-world quantity cannot literally be a complex number, the math describing it often produces complex intermediate results.
Is this the same as completing the square?
Yes — the quadratic formula is exactly what you get by completing the square on the general equation ax² + bx + c = 0 once and for all, so you never have to redo that algebra by hand.
Updated
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