Solve any quadratic equation with the quadratic formula. Find both roots, discriminant, vertex, axis of symmetry, and factored form with step-by-step solutions.
The quadratic formula is one of the most important tools in algebra: x = (−b ± √(b² − 4ac)) / 2a. It provides an exact solution for any equation of the form ax² + bx + c = 0, regardless of whether the roots are real, repeated, or complex. Our quadratic formula calculator lets you enter coefficients a, b, and c, then instantly displays both roots, the discriminant value, vertex coordinates, axis of symmetry, y-intercept, and factored form when integer factors exist.
Understanding the discriminant (Δ = b² − 4ac) is key to predicting the nature of the solutions before you even compute them. When Δ > 0 you get two distinct real roots, when Δ = 0 you get one repeated root, and when Δ < 0 the roots are complex conjugates. This calculator color-codes the result so you can immediately see which case applies.
Beyond finding roots, the calculator reports parabola properties: whether the curve opens upward or downward, the minimum or maximum vertex point, and the steepness determined by the leading coefficient. A step-by-step solution table walks through each computation so students can follow the algebra and verify their own work. Eight common equation presets let you instantly load classic textbook problems for quick exploration.
Quadratic Formula Calculator — Solve ax² + bx + c = 0 helps you solve quadratic formula calculator — solve ax² + bx + c = 0 problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Coefficient a, Coefficient b, Coefficient c once and immediately inspect Root x₁, Root x₂, Discriminant (Δ) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
x = (−b ± √(b² − 4ac)) / 2a, where Δ = b² − 4ac is the discriminant. Vertex at (−b/2a, f(−b/2a)). Sum of roots = −b/a. Product of roots = c/a.
Result: Root x₁ shown by the calculator
Using the preset "x²−5x+6=0", the calculator evaluates the quadratic formula calculator — solve ax² + bx + c = 0 setup, applies the selected algebra rules, and reports Root x₁ with supporting checks so you can verify each transformation.
This calculator takes Coefficient a, Coefficient b, Coefficient c and applies the relevant quadratic formula calculator — solve ax² + bx + c = 0 relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Root x₁, Root x₂, Discriminant (Δ), Vertex to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
It is x = (−b ± √(b² − 4ac)) / 2a, used to solve any second-degree polynomial equation ax² + bx + c = 0. Use this as a practical reminder before finalizing the result.
If Δ > 0 there are two distinct real roots; Δ = 0 gives one repeated root; Δ < 0 means two complex conjugate roots. Keep this note short and outcome-focused for reuse.
Yes. When the discriminant is negative, both roots are displayed in a + bi form.
The vertex is the highest or lowest point on the curve, located at x = −b/2a. If a > 0 it is a minimum; if a < 0 it is a maximum.
A quadratic with integer coefficients factors neatly when the discriminant is a perfect square. This calculator shows the factored form when it exists.
Vieta's formulas state that for ax² + bx + c = 0, the sum of roots equals −b/a and the product of roots equals c/a, which can be used to check your answers. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.