Quadratic Formula Calculator
Apply the quadratic formula x = (-b ± √(b² - 4ac)) / 2a to find exact solutions.
Works for any coefficients with complete step-by-step explanation.
Quadratic Equation Solver
Our free quadratic equation solver helps you find the roots of any quadratic equation using the quadratic formula. Get the discriminant, vertex form, axis of symmetry, factored form, and complete step-by-step solutions. Works with real and complex roots.
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Updated: July 2026
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Quadratic Calculator calculator
Equation
1x² - 5x + 6 = 0
Discriminant (Δ)
1
2 Real Roots
Δ = (-5)² - 4(1)(6) = 25 - 24 = 1
Quadratic Formula
x = (-b ± √Δ) / 2a
Solutions
x₁
3
x₂
2
Properties
Vertex (h, k)
(2.5, -0.25)
Axis of Symmetry
x = 2.5
Sum of Roots
5
Product of Roots
6
Forms
Factored:
(x - 3)(x - 2)
Vertex:
(x - 2.5)² - 0.25
Solution Steps
1. Δ = b² - 4ac = 25 - 24 = 1
2. x = (-(-5) ± √1) / 2(1)
3. x = (5 ± 1) / 2
4. x₁ = 3, x₂ = 2
lightbulb Tips
- •x = (-b ± √(b²-4ac)) / 2a
- •Δ > 0: two real roots
- •Δ = 0: one repeated root
- •Δ < 0: complex roots
functions Reference
Discriminant (Δ = b²-4ac)
Δ > 0
Two real roots
Δ = 0
One repeated root
Δ < 0
Complex roots
Formulas
Vertex: (-b/2a, f(-b/2a))
Sum of roots: -b/a
Product of roots: c/a
How to Use the Quadratic Calculator
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Enter Coefficients
Input values for a, b, and c in ax² + bx + c = 0.
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View Roots
See the solutions x₁ and x₂ (real or complex).
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Analyze Equation
Get discriminant, vertex, axis of symmetry, and forms.
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Follow Steps
Learn the complete solution process step by step.
The Formula
The quadratic formula gives the solutions to any quadratic equation. The discriminant (Δ) determines whether roots are real (Δ ≥ 0) or complex (Δ < 0).
x = (-b ± √(b² - 4ac)) / 2a
lightbulb Variables Explained
- a, b, c Coefficients in ax² + bx + c = 0
- Δ = b² - 4ac Discriminant (determines nature of roots)
- x₁, x₂ The two roots (solutions)
tips_and_updates Pro Tips
1
If Δ > 0: Two distinct real roots
2
If Δ = 0: One repeated real root (perfect square)
3
If Δ < 0: Two complex conjugate roots
4
Vertex form: a(x - h)² + k where h = -b/2a
5
Sum of roots = -b/a, Product of roots = c/a
6
The parabola opens upward if a > 0, downward if a < 0
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Get real or complex roots, discriminant, vertex, axis of symmetry, and detailed step-by-step solutions.
Discriminant Calculator
Calculate the discriminant Δ = b² - 4ac to determine the nature of roots before solving.
- Positive means two real roots
- zero means one root
- negative means complex roots
Vertex Form Calculator
Convert standard form to vertex form a(x - h)² + k.
Find the vertex coordinates and understand the parabola's position and direction.
Factoring Quadratic Calculator
Get the factored form a(x - r₁)(x - r₂) when roots are found.
Because roots often turn out to be fractions rather than whole numbers, a fraction calculator makes it easy to simplify them, and the factored form is useful for understanding the equation's structure and solving inequalities.
Complex Roots Solver
When the discriminant is negative, find complex conjugate roots in the form a ± bi.
Our solver handles both real and imaginary solutions.
Step-by-Step Solutions
Learn how to solve quadratic equations with detailed explanations.
Perfect for students studying algebra and preparing for exams.
What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
Because the highest power of the unknown x is two, the equation has at most two solutions, called roots. Geometrically, its graph is a parabola that opens upward when a > 0 and downward when a < 0.
According to Wolfram MathWorld, the term "quadratic" derives from the Latin quadratus, meaning square. Quadratics appear across algebra, physics, and finance, making them one of the most studied equation types in elementary mathematics.
How to Solve a Quadratic Equation Using the Quadratic Formula
To solve a quadratic equation, apply the quadratic formula x = (-b ± √(b² - 4ac)) / 2a after writing the equation in standard form and identifying a, b, and c.
First compute the discriminant b² - 4ac, then substitute into the formula and evaluate both the plus and minus cases. For example, 2x² - 4x - 6 = 0 gives discriminant 16 + 48 = 64, so x = (4 ± 8) / 4, yielding x = 3 and x = -1.
Khan Academy notes this formula works for every quadratic, unlike factoring or completing the square, which are situational. The formula is derived by completing the square on the general form.
Completing the Square Method Explained
Completing the square rewrites a quadratic as a perfect-square trinomial plus a constant, converting ax² + bx + c = 0 into the vertex form a(x - h)² + k = 0.
For x² + 6x + 5 = 0, add and subtract (6/2)² = 9 to get (x + 3)² - 4 = 0, so (x + 3)² = 4 and x = -1 or x = -5.
Encyclopaedia Britannica describes this classical technique as the basis for deriving the quadratic formula itself. Beyond solving, completing the square reveals the parabola's vertex directly, making it valuable for graphing and for integrating rational functions in calculus.
Real-World Uses of Quadratic Equations
Quadratic equations model many real-world situations involving area, motion, and optimization.
- In physics, projectile height follows h = -½gt² + v₀t + h₀, a quadratic in time whose roots reveal when an object hits the ground.
- Engineers use them to size rectangular areas.
- Economists use them to find profit-maximizing prices where revenue is a parabola.
According to the NIST Digital Library of Mathematical Functions, polynomial roots underpin numerical analysis and signal processing. Wolfram MathWorld highlights that the parabola's symmetry, captured by the axis x = -b/(2a), makes quadratics ideal for describing reflective dishes, headlights, and satellite antennas.
Common Mistakes When Solving Quadratic Equations
The most common mistake is applying the quadratic formula before rewriting the equation in standard form ax² + bx + c = 0, so all terms sit on one side and equal zero.
- Another frequent error is mishandling signs: b in x² - 5x + 6 is -5, not 5, which changes -b to +5.
- Students also forget the ± symbol and report only one root, or drop the 2a denominator under both terms.
- When the discriminant is negative, do not conclude "no solution" — the roots are complex conjugates, as Khan Academy explains.
Finally, always verify roots by substituting them back into the original equation.
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