Quadratic Formula Calculator
Formula:
Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
Discriminant: Δ = b² - 4ac
Vertex: (-b/2a, f(-b/2a))
Axis of Symmetry: x = -b/2a
Nature of roots:
- If Δ> 0: Two distinct real roots
- If Δ = 0: One repeated real root
- If Δ < 0: Two complex conjugate roots
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What Your Result Means
- Discriminant (Δ): Determines whether roots are real or complex. Δ = b² − 4ac.
- Root 1 & Root 2 (x₁, x₂): The solutions to the quadratic equation (where the parabola crosses the x-axis).
- Vertex: The minimum or maximum point of the parabola.
- Axis of Symmetry: The vertical line through the vertex that divides the parabola into two mirror halves.
How This Calculator Works
This calculator applies the quadratic formula to solve equations of the form ax² + bx + c = 0. It computes the discriminant (Δ = b² − 4ac) to determine if roots are real or complex, then calculates both roots using x = (−b ± √Δ) / 2a. It also finds the vertex coordinates (−b/2a, f(−b/2a)) and the axis of symmetry x = −b/2a. Results update instantly as you type.
Quick Questions
What if a = 0?
The equation is not quadratic if a = 0 (it becomes linear). The calculator will alert you and skip the calculation.
What are complex roots?
Complex roots occur when Δ < 0. They are expressed as real ± imaginary parts (e.g., 2 + 3i). The parabola does not cross the x-axis.
What's the difference between roots and vertex?
Roots are where the parabola crosses the x-axis (y = 0). The vertex is the peak or trough of the parabola, found at x = −b/2a.
Why are there two roots?
Quadratic equations are degree 2, so they always have exactly 2 roots (real, repeated, or complex). The parabola can intersect the x-axis at 0, 1, or 2 points.
Sources
- Britannica: Quadratic Equation (Definition and history of quadratic formulas)
- Wolfram MathWorld: Quadratic Formula (Mathematical reference and proof)
Method & review
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.