Quadratic Formula Calculator
Calculated using standard mathematical and industry formulas.
Free online Quadratic Formula Calculator calculator and unit conversion tool for everyday, engineering, and home projects.
How to use this calculator?
- Enter coefficient a (the multiplier of x²).
- Enter coefficient b (the multiplier of x).
- Enter constant c.
- Read your primary root (x₁), secondary root (x₂), discriminant Δ (b² - 4ac), and parabola vertex (h, k).
Formula & calculation method
Quadratic Formula & Discriminant Equations
x = [ −b ± √(b² − 4ac) ] / (2a) · Discriminant Δ = b² − 4ac · Vertex (h, k) = ( −b/(2a) , c − b²/(4a) )
a ≠ 0. If Δ < 0, roots are expressed in standard complex form a + bi.
Step-by-step calculation example
x² − 5x + 6 = 0 (a=1, b=-5, c=6) → Δ = 25 − 24 = 1 > 0 → x₁ = (5+1)/2 = 3, x₂ = (5-1)/2 = 2. Vertex: (2.5, -0.25).
Key insights to remember
- The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 yields 2 distinct real roots, Δ = 0 yields 1 repeated real root, and Δ < 0 yields 2 complex conjugate roots.
- The vertex of the quadratic parabola ax² + bx + c occurs at x = −b / (2a); if a > 0 the parabola opens upwards, if a < 0 it opens downwards.
- Quadratic equations model physical phenomena including projectile motion, acceleration, satellite dish geometry, and profit optimization curves.
- By Vieta's formulas, the sum of the roots x₁ + x₂ = −b/a and the product x₁ × x₂ = c/a.
How to interpret your results?
Roots x₁ and x₂
The x-intercepts where the quadratic graph intersects the horizontal axis (ax² + bx + c = 0).
Discriminant (Δ)
Indicates whether the parabola crosses the x-axis twice (Δ > 0), touches it at a single point (Δ = 0), or hovers above/below without touching (Δ < 0).
Parabola Vertex (h, k)
The minimum point (if a > 0) or maximum peak (if a < 0) of the parabolic function.
Frequently asked questions
- How do I use the Quadratic Formula Calculator calculator?
- Enter your numerical input and select desired units to get instant accurate results.
- How accurate are the conversions?
- Conversions use official ISO conversion factors for maximum precision.
- Does it convert metric and imperial units?
- Yes, seamless conversions between metric (meters, kg) and imperial (feet, lbs) are supported.
- Is my data stored?
- No, all calculations execute 100% locally in your browser with zero data logging.
- Can I use this for DIY or engineering projects?
- Yes, it provides reliable estimates for construction, crafting, science, and everyday math.
- What if I enter invalid inputs?
- The calculator will display a friendly error prompt if non-numeric values are entered.
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