Quadratic Equation Calculator (Quadratic Formula, Δ Cases)
Solve ax² + bx + c = 0 with full discriminant analysis. Returns real roots, complex roots, parabola vertex, and step-by-step solution for educational use.
Based on 2 references↓Discriminant (Δ): Δ = 1
Δ > 0 → two distinct real roots
Step-by-step (quadratic formula)
- Discriminante Δ = b² − 4ac = (-5)² − 4·(1)·(6) = 1
- Δ > 0 → duas raízes reais distintas
- x = (−b ± √Δ) / 2a = (5 ± √1) / 2 = (5 ± 1) / 2
- x₁ = 2, x₂ = 3
Parabola vertex
(2.5, -0.25)
Y-intercept
y = 6
The quadratic formula (Bhaskara, in BR schools) solves any equation of the form ax² + bx + c = 0 by closed-form expression: x = (−b ± √Δ) / 2a, where Δ = b² − 4ac is the discriminant. The discriminant tells you, before solving, what kind of solution to expect: Δ > 0 means two distinct real roots (the parabola crosses the x-axis at two points), Δ = 0 means one real double root (the parabola is tangent to the x-axis), Δ < 0 means two complex conjugate roots (the parabola does not touch the x-axis).
BR schools formalise this as "fórmula de Bhaskara" and it is part of every vestibular exam (ENEM, FUVEST, UNICAMP). EN classrooms call it "the quadratic formula". Same math, different naming. The calculator returns step-by-step solution suitable for showing your work + the parabola vertex (h, k) for graphing context, since many problems ask for vertex form alongside roots.
The discriminant decides the case
Discriminant Δ = b² − 4ac. Compute this first. Sign tells you everything:
Δ > 0 → two distinct real roots. x₁ = (−b − √Δ) / 2a, x₂ = (−b + √Δ) / 2a. Parabola crosses the x-axis at two points (the roots).
Δ = 0 → one real double root. x = −b / 2a. Parabola is tangent to the x-axis at that single point.
Δ < 0 → two complex conjugate roots. x = (−b ± i·√(−Δ)) / 2a. Parabola does not cross the x-axis. Roots come in conjugate pairs because i² = −1 — the imaginary parts have opposite signs.
Vertex (h, k): the parabola's extremum. h = −b / 2a (same as the double-root formula). k = c − b² / 4a (the y-value at h). For graphing or completing-the-square form: y = a(x − h)² + k.
Practical examples
Δ > 0: x² − 5x + 6 = 0
Setup: a=1, b=-5, c=6.
Δ = 25 − 24 = 1 (positive). x = (5 ± 1) / 2 = **{2, 3}**. Vertex (2.5, -0.25).
Takeaway: Two distinct real roots. Factored form: (x − 2)(x − 3) = 0. Most school problems are this case.
Δ = 0: x² − 4x + 4 = 0
Setup: a=1, b=-4, c=4.
Δ = 16 − 16 = 0. x = 4 / 2 = **2** (double root). Vertex (2, 0).
Takeaway: Perfect square: (x − 2)² = 0. The parabola is tangent to the x-axis at x=2.
Δ < 0: x² + x + 1 = 0
Setup: a=1, b=1, c=1.
Δ = 1 − 4 = -3 (negative). x = (-1 ± i·√3) / 2 = **-0.5 ± 0.866i**. Vertex (-0.5, 0.75).
Takeaway: Complex conjugate pair. The parabola does not cross the x-axis (vertex above it).
Vertex form for graphing: 2x² − 8x + 6
Setup: a=2, b=-8, c=6. Find roots + vertex.
Δ = 64 − 48 = 16. x = (8 ± 4) / 4 = **{1, 3}**. Vertex h = 8/4 = 2, k = 6 − 64/8 = -2. Vertex: (2, -2).
Takeaway: Vertex form y = 2(x − 2)² − 2 — useful for graphing without plotting points.
Frequently asked questions
Why does a = 0 return null?▾
When a = 0, the equation reduces to bx + c = 0 (linear, not quadratic). The quadratic formula has 2a in the denominator and divides by zero. For linear equations, x = −c / b. Use a different solver.
What is i (imaginary unit)?▾
i = √-1 by definition. Allows representing solutions to equations with no real solution, e.g. x² + 1 = 0 has roots x = ±i. Real numbers are extended to complex numbers (a + bi) for full algebraic closure. The calc surfaces complex roots when Δ < 0.
What is "Bhaskara"?▾
In Brazilian Portuguese, the quadratic formula is named after Bhaskara II (12th-century Indian mathematician), even though the formula itself was used by various civilizations earlier. Outside BR, the formula is just "quadratic formula" or sometimes "Bhāskara's method".
How do I factor a quadratic without the formula?▾
Factor by inspection: ax² + bx + c = a(x − r₁)(x − r₂) where r₁ + r₂ = -b/a and r₁ × r₂ = c/a (Vieta's formulas). Faster than the formula when roots are integers. The formula is the universal fallback.
Why does the calc show vertex coordinates?▾
Many quadratic problems ask for vertex (max or min) alongside roots. Physics: maximum height of a projectile. Business: maximum profit at quantity h. The vertex (h, k) is the parabola's extremum: h = -b/2a, k = c - b²/4a.
Sources & references
Cross-check every number in this calculator against the primary sources below.
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