Equation
x² - 5x + 6 = 0
Two distinct real roots
The equation has two real solutions: x = 2 and x = 3.
Root 1
x = 2
Root 2
x = 3
Discriminant
1
Axis
x = 2.5
Vertex
(2.5, -0.25)
Factored form
(x - 2)(x - 3) = 0
Free quadratic equation solver
Quadratic Formula Calculator
A quadratic formula calculator solves equations in the form ax² + bx + c = 0. Enter the coefficients to see real or complex roots, the discriminant, vertex, axis of symmetry, and clear step-by-step work.
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Step-by-step solution
- 1
Identify a = 1, b = -5, and c = 6.
- 2
Find the discriminant: b² - 4ac = (-5)² - 4(1)(6) = 1.
- 3
Because the discriminant is positive, the equation has two real roots.
- 4
x = (-b ± √D) / 2a = (5 ± √1) / 2.
- 5
The roots are x = 2 and x = 3.
How to use the quadratic formula calculator
Enter a, b, and c
Match your equation to ax² + bx + c = 0 and type each coefficient.
Calculate the discriminant
The tool evaluates b² - 4ac to classify the roots.
Read the solutions
Review the roots, vertex, axis of symmetry, and factored form when available.
Follow the steps
Use the substitution and simplification steps to check your work.
Quadratic formula FAQ
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves any equation in the form ax² + bx + c = 0 when a is not zero.
What does the discriminant tell you?
The discriminant is b² - 4ac. A positive value gives two real roots, zero gives one repeated real root, and a negative value gives two complex-conjugate roots.
Can this calculator solve complex roots?
Yes. When the discriminant is negative, the calculator writes both answers with i, the imaginary unit, and shows the real and imaginary parts.
What happens when a equals zero?
When a is zero, the equation is not quadratic. The calculator automatically solves the remaining linear equation bx + c = 0 or explains why it has no unique solution.
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