Quadratic Formula Calculator
Solve ax² + bx + c = 0 with step-by-step solutions, discriminant analysis, and parabola details.
Enter Coefficients
1x² − 5x + 6 = 0
x = (−b ± √(b²−4ac)) / (2a)
Results
Root Type
Two distinct real roots
Discriminant (Δ)
1
Y-Intercept
(0, 6)
x₁
3
x₂
2
Vertex
(2.5, -0.25)
Axis of Symmetry
x = 2.5
Factored Form
(x − 3)(x − 2)
Parabola
opens upward ∪ (minimum)
Step-by-Step Solution
Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 with full step-by-step workings, discriminant analysis, vertex, and factored form.
Guide
How it works
A 4,000-Year-Old Formula
Quadratic equations have been solved since at least 2000 BC. Babylonian mathematicians solved equivalent problems using geometric methods — completing the square on clay tablets, without algebraic notation. Ancient Indian and Greek mathematicians developed similar techniques. The explicit formula we use today was popularized by the Persian mathematician al-Khwarizmiaround 820 AD in his book "Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala" — the origin of the word algebra.
The Parabola and Its Roots
Every quadratic equation ax² + bx + c = 0 corresponds to a parabola y = ax² + bx + c. The roots (solutions) are the x-coordinates where this parabola crosses the x-axis. If the discriminant Δ = b² − 4ac is positive, the parabola crosses the x-axis twice (two real roots). If Δ = 0, it just touches the axis (one repeated root, vertex on x-axis). If Δ < 0, it never crosses (complex roots, parabola entirely above or below the x-axis).
Deriving the Formula: Completing the Square
Starting from ax² + bx + c = 0, divide by a: x² + (b/a)x + c/a = 0. Move c/a to the right: x² + (b/a)x = −c/a. Add (b/2a)² to both sides: (x + b/2a)² = (b² − 4ac)/(4a²). Take the square root and solve: x = (−b ± √(b² − 4ac)) / (2a). This algebraic manipulation is called completing the square and reveals why the vertex x-coordinate is −b/(2a).
Three Forms of a Quadratic
Quadratics appear in three equivalent forms: Standard form (ax² + bx + c) shows y-intercept directly. Vertex form (a(x − h)² + k) reveals the vertex (h, k) at a glance. Factored form (a(x − r₁)(x − r₂)) shows roots directly. Converting between forms is a core algebra skill.
Vieta's Formulas
French mathematician François Viète discovered elegant relationships between roots and coefficients: x₁ + x₂ = −b/a and x₁ × x₂ = c/a. These allow you to check your answers without substituting back into the equation, and are used to construct polynomials with desired roots.
Real-World Applications
Projectile motion: The height h = −16t² + v₀t + h₀ (feet, imperial) is quadratic in time t. Solving for h = 0 gives when the projectile lands. Profit maximization:Revenue minus cost often yields a quadratic in price or quantity. The vertex gives the optimal price. Engineering: Cable suspension bridges form parabolas under uniform load. Optics: Parabolic mirrors focus parallel light rays to a single focal point, used in telescopes and satellite dishes.
What does it mean when the discriminant is negative?expand_more
A negative discriminant means the quadratic has no real solutions — only complex (imaginary) roots. The parabola doesn't cross the x-axis. In physics, this might mean a projectile never reaches a certain height; in engineering, it might indicate a system never reaches equilibrium under given conditions.
Can a = 0 in a quadratic equation?expand_more
No. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0 because we divide by 2a during the derivation. A linear equation has exactly one solution: x = −c/b.
What are complex roots and when do they appear?expand_more
Complex roots appear when Δ < 0 and take the form p ± qi, where i = √(−1). They always come in conjugate pairs. While they have no geometric meaning on the real number line, complex roots are essential in electrical engineering (AC circuits), signal processing, and quantum mechanics.
How do I check if my roots are correct?expand_more
Substitute each root back into ax² + bx + c. The result should equal 0. You can also use Vieta's formulas: check that x₁ + x₂ = −b/a and x₁ × x₂ = c/a. Both checks should hold simultaneously for correct roots.
Why is the vertex at x = −b/(2a)?expand_more
The vertex occurs at the axis of symmetry, which is exactly halfway between the two roots. Using Vieta's formulas: (x₁ + x₂)/2 = (−b/a)/2 = −b/(2a). It's also the point where the derivative of ax² + bx + c equals zero, confirming it's a maximum or minimum.