Quadratic Equation Solver
Solve quadratic equations (ax² + bx + c = 0) with the quadratic formula. Shows the discriminant Δ and real roots.
Enter the coefficients of the equation ax² + bx + c = 0
Quadratic Formula
For the equation ax² + bx + c = 0 (with a ≠ 0), you go through the discriminant (Δ) first and then the formula below:
- Δ = b² − 4ac
- Δ > 0 → dois valores reais distintos
- Δ = 0 → one real root (double root)
- Δ < 0 → sem raízes reais
Quadratic (second-degree) equations
A second-degree equation has the form ax² + bx + c = 0 with a ≠ 0. The quadratic formula (often called Bhaskara's formula in Brazil) gives the roots: x = (-b ± √Δ)/(2a) with discriminant Δ = b² - 4ac. The sign of Δ tells how many real roots there are: Δ > 0 two distinct real roots; Δ = 0 one double real root; Δ < 0 two complex conjugate roots.
By Girard's relations, the sum of the roots is S = -b/a and their product is P = c/a. Completing the square yields the vertex form a(x - h)² + k, where the vertex of the parabola is (h, k) = (-b/(2a), -Δ/(4a)).
Applications and history
Quadratics model projectile motion (trajectory under constant gravity), parabolic optics (headlight reflectors, satellite dishes), area optimisation and many physics and engineering problems. The popular attribution to Bhaskara II (12th century India) is partially apocryphal: the same formula already appeared in Brahmagupta (628) and in Al-Khwarizmi (~820), who solved quadratics geometrically. Bhaskara II made other contributions to algebra and to negative roots; the "Bhaskara formula" label is a Brazilian school tradition.
FAQ
What if a = 0? The equation becomes linear (bx + c = 0) and the quadratic formula no longer applies — divide by b instead.
What does the discriminant mean geometrically? The parabola y = ax² + bx + c crosses the x-axis at the real roots; Δ > 0 means two crossings, Δ = 0 means it touches at one point, Δ < 0 means no crossing.
How do I factor a quadratic? If r₁ and r₂ are the roots, then ax² + bx + c = a(x - r₁)(x - r₂).
Can I avoid the discriminant? Yes — completing the square or using Girard's relations with the vertex form works too, but Δ is usually faster to compute.
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Solve quadratic equations step by step
The quadratic equation, written as ax² + bx + c = 0, is one of the pillars of school maths, and the quadratic formula (Bhaskara, as it is known in Brazil) is the classic way to solve it. This tool applies the formula in your place. You enter the coefficients a, b and c, and it hands back the roots.
Beyond the solutions, it shows the discriminant, the famous Δ, which gives away the story before the roots even appear. Positive means two distinct real roots, zero indicates a double root, and negative points to no real root at all. Looking at the delta helps you understand the result instead of just memorising it, which comes in handy when the goal is exam prep.
The calculation happens in the browser itself. Swap any coefficient and the roots recompute, so checking exercises and trying out different equations turns out to be easy.