Math

Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0 and see the discriminant, both roots and the vertex.

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Discriminant (b² − 4ac)
Discriminant (b² − 4ac)
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Roots
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Vertex x
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Vertex y
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Quadratic Equation Solver by TheCalcTools

How the Quadratic Equation Solver works

Enter the three coefficients.

The formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Worked example

Using the default values shown in the calculator:

a (x² coefficient)1
b (x coefficient)-3
c (constant)2
  1. Discriminant = (−3)² − 4×1×2 = 1.00.
  2. x = 2.0000 and x = 1.0000.

Result:

Discriminant (b² − 4ac)1.00
Rootsx = 2.0000 and x = 1.0000
Vertex x1.5000
Vertex y-0.2500

Frequently asked questions

What does the discriminant tell me?

Positive: two real roots. Zero: one repeated root. Negative: two complex roots.

Sources and further reading

Linear Equation SolverSolve a one-variable linear equation of the form ax + b = c System of Equations Calculator (2×2)Solve a system of two linear equations, a₁x + b₁y = c₁ and a₂x + b₂y = c₂ Cubic Equation Calculator (Solve ax³ + bx² + cx + d = 0)Enter the four coefficients to find every root Complex Number Calculator (Add, Multiply, Divide, Polar Form)Enter two complex numbers a + bi and c + di and choose the operation Exponential Growth and Decay CalculatorEnter the starting value, rate and time Parabola Calculator (Vertex, Focus, Directrix, Vertex Form)Enter the coefficients a, b and c

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