Quadratic equation calculator
Solve ax² + bx + c = 0 with the discriminant, both roots and the parabola vertex.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Solve ax² + bx + c = 0 with the discriminant, both roots and the parabola vertex.
- Formula used
ax² + bx + c = 0- Example
- x² − 5x + 6 = 0
Quadratic equation calculator
The quadratic formula solves any second-degree equation, but it is the sign of the discriminant that decides what kind of solutions you will get.
Beyond the roots, the calculator reports the vertex, which is what you need to sketch the parabola.
How it works
A positive discriminant gives two distinct real roots; zero gives one repeated root; a negative value gives two complex conjugate roots, shown with the imaginary unit i.
Formula
ax² + bx + c = 0
Δ = b² − 4ac
x = (−b ± √Δ) / (2a)
Vertex: x = −b / (2a)
Worked example
x² − 5x + 6 = 0
Δ = 25 − 24 = 1
x₁ = (5 + 1) / 2 = 3
x₂ = (5 − 1) / 2 = 2
Vertex: x = 2.5 ; y = −0.25
Explanation
The discriminant tells you everything before you solve
The quantity under the root, b² minus 4ac, decides the nature of the solutions without needing to calculate them. If it is positive there are two distinct real solutions; if it is zero, a single repeated solution; if negative, two complex conjugate solutions. In an exam, checking it first saves work.
What it means graphically
A quadratic equation traces a parabola. The solutions are the points where it crosses the horizontal axis. A positive discriminant means the parabola crosses the axis twice; zero, that it touches at the vertex; negative, that it passes entirely above or below without touching.
Where it appears outside the classroom
The trajectory of a projectile, the fall time of an object, the point where two cars meet, the calculation of an area when the perimeter and a relationship between sides are known. All end in an equation of this kind.
Vieta's formulas
The sum of the two solutions is minus b over a, and their product is c over a. It is a quick way to check a result: if the sum and product do not match, there is an error somewhere.
Frequently asked questions
Why can a not be zero?
Because the squared term would disappear and the equation would become first-degree.
What are complex roots?
When the discriminant is negative the parabola never crosses the X axis. The roots still exist but involve the imaginary unit i = √−1.
How do I check my solutions are right?
Their sum should equal minus b over a, and their product c over a. If both hold, the result is consistent.
What does the discriminant mean on the graph?
How many times the parabola crosses the horizontal axis: twice if positive, once if zero and never if negative.
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