Quadratic Equation Calculator
Enter the three coefficients of ax² + bx + c = 0 and the quadratic formula does the rest. The discriminant b² − 4ac decides the answer: positive gives two real roots, zero gives one repeated root, and negative gives a conjugate pair of complex roots.
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Roots
x = 4 or x = -1
Nature of the roots
Two distinct real roots
Details
Updates as you typeMust not be 0 — with a = 0 the equation is linear, not quadratic.
Summary
Roots
x = 4 or x = -1
- Discriminant (b² − 4ac)
- 25.0000
- Nature of the roots
- Two distinct real roots
- First root
- x = 4
- Second root
- x = -1
- Real roots
- 2
- Vertex x
- 1.5000
- Vertex y
- -6.2500
- Axis of symmetry
- x = 1.5
How this is calculated
- a
- 1.0000
- b
- -3.0000
- c
- -4.0000
- b²
- 9.0000
- 4ac
- -16.0000
- Discriminant b² − 4ac
- 25.0000
- −b ÷ 2a (vertex x)
- 1.5000
- Minimum or maximum value
- -6.2500
- Opens
- Upward (vertex is the minimum)
- Sum of roots (−b ÷ a)
- 3.0000
- Product of roots (c ÷ a)
- -4.0000
- Roots
- x = 4 or x = -1
Compare scenarios
See how one change moves the result
- CurrentYour inputs as they standx = 4 or x = -1Current
- a (x² coefficient)1.3x = 3.253438 or x = -0.945745
- b (x coefficient)-3.7x = 4.574427 or x = -0.874427
- c (constant term)-5x = 4.192582 or x = -1.192582
Frequently asked questions
What is the quadratic formula?
For ax² + bx + c = 0 with a not equal to 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ (2a). The expression under the square root, b² − 4ac, is called the discriminant and it determines how many real solutions exist.
What does the discriminant tell you?
If b² − 4ac is positive the parabola crosses the x-axis twice, giving two distinct real roots. If it is exactly zero the parabola touches the axis at one point, a repeated root. If it is negative there is no real solution and the two roots form a complex conjugate pair.
Why must a not be zero?
With a = 0 the x² term disappears and bx + c = 0 is a linear equation, not a quadratic. The quadratic formula also divides by 2a, so a zero leading coefficient makes it undefined. This calculator reports an error instead of returning a meaningless number.
How do you find the vertex of a parabola?
The vertex sits on the axis of symmetry at x = −b ÷ (2a); substitute that back into the equation to get the y value. When a is positive the vertex is the lowest point of the curve, and when a is negative it is the highest point.
What are complex roots written like?
They are written in the form p + qi, where i is the square root of −1. The two roots always come as a matching pair such as −1 + 2i and −1 − 2i, sharing the same real part −b ÷ (2a) and equal but opposite imaginary parts.