Quadratic Formula Calculator

How to use the Quadratic Formula Calculator

What this calculator does

This calculator solves any valid second-degree equation that can be represented as Ax² + Bx + C = 0 with real coefficients and A not equal to zero. It computes the discriminant, classifies the roots, returns real or complex solutions, locates the vertex and axis of symmetry, shows equivalent equation forms, draws the parabola, and creates a structured Excel workbook. It is a numerical and algebraic aid; it does not prove that a real-world model is appropriate or that measured coefficients are accurate.

When to use it

Use it to check homework or hand calculations, convert a quadratic between common forms, inspect whether a model crosses the x-axis, or identify the maximum or minimum point of a parabolic relationship. It is also useful when reviewing projectile-height models, area optimization problems, revenue or cost curves, and other situations that reduce to a quadratic equation.

How to calculate

1. The calculator opens with the complete demonstration equation x² – 3x + 2 = 0. Its results, graph, table, and example Excel workbook are immediately available.
2. Choose Formula form. Standard form uses A, B, and C; vertex form uses A, H, and K; factored form uses A, x₁, and x₂. Switching forms converts a currently valid equation, so the underlying function stays the same.
3. Replace the sample values with your coefficients. Plain decimal notation and correctly grouped thousands are accepted; scientific notation and ambiguous decimal-comma entries are rejected.
4. Read Solutions first, then use Discriminant (Δ), Vertex, Axis of symmetry, and Y-intercept to interpret the curve. The equivalent-form cards, graph, and calculation table all come from the same model.
5. Select Download Excel to export the current valid state as a real .xlsx workbook. Reset clears the demonstration and all entered coefficients; export is then disabled until a complete valid equation is entered again.

Input guide

Formula form is required and chooses the coefficient system. In Standard form, A – quadratic coefficient is a required nonzero real number, while B – linear coefficient and C – constant term are required finite real numbers. For example, A = 1, B = – 3, and C = 2 produce x² – 3x + 2. A common mistake is setting A to zero, which makes the equation linear rather than quadratic.

In Vertex form, A – vertical scale is required and nonzero, H – vertex x-coordinate is the horizontal location of the turning point, and K – vertex y-coordinate is its vertical location. For example, A = 1, H = 1.5, and K = – 0.25 represent the startup equation. Increasing |A| makes the parabola narrower; changing the sign of A reverses its opening direction. Do not confuse the sign inside A(x – H)² + K: a visible “x – 3” means H = 3, not – 3.

In Factored form, A – vertical scale is required and nonzero, while x₁ – first real root and x₂ – second real root are required real numbers. For example, A = 1, x₁ = 2, and x₂ = 1 produce (x – 2)(x – 1). Repeated values are allowed and create a double root. This input form represents only real factors; equations with complex roots can still be entered in standard or vertex form.

Output guide

Solutions gives the two roots, one repeated root, or a conjugate complex pair. Discriminant (Δ) is B² – 4AC: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots. Vertex is the minimum point when A is positive and the maximum point when A is negative. Axis of symmetry is the vertical line through that vertex. Y-intercept is (0, C). The Standard form, Vertex form, and Factored form cards are exact algebraic identities subject only to display rounding; factored form is reported as unavailable over the reals when the roots are complex.

The Parabola and key points graph plots the current function across a range centered on the vertex. The blue curve represents f(x), the teal point is the vertex, and purple points are real roots. The Calculation steps table lists the standard equation, B², 4AC, Δ, and the final quadratic-formula substitution. These are calculations, not recommendations.

Worked example

For A = 1, B = – 3, and C = 2, the discriminant is ( – 3)² – 4(1)(2) = 9 – 8 = 1. The quadratic formula gives x = (3 ± √1) / 2, so x₁ = 2 and x₂ = 1. The vertex is at H = – B/(2A) = 1.5, and K = f(1.5) = – 0.25. Those values match the first-open result cards, graph markers, calculation table, and workbook checkpoints.

Learn more

For a textbook treatment of the formula and the role of the discriminant, see the OpenStax section on quadratic equations. For graph interpretation, including vertices and opening direction, review OpenStax on quadratic functions.