Quadratic Equation Calculator
Enter coefficients for an equation in the form ax² + bx + c = 0.
This calculator returns real or complex roots, discriminant, axis of symmetry, and vertex.
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The highest power of x is 2, which makes the graph of the equation a parabola.
This quadratic formula calculator helps you quickly solve equations, check homework, and understand how each coefficient changes the roots. It works for:
- Two distinct real roots
- One repeated real root
- Two complex conjugate roots
How the Quadratic Formula Calculator Works
The calculator uses the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
The expression inside the square root, b² - 4ac, is called the discriminant. It determines the type of roots:
- Discriminant > 0: two different real roots
- Discriminant = 0: one real root (double root)
- Discriminant < 0: two complex roots
Step-by-Step Manual Method (If You Want to Check by Hand)
1) Identify coefficients
From ax² + bx + c = 0, read off a, b, and c carefully. Sign errors are the most common mistake.
2) Compute discriminant
Calculate D = b² - 4ac. Keep parentheses around negative numbers.
3) Substitute into formula
Plug values into x = (-b ± √D) / (2a), simplify, and reduce if possible.
4) Interpret the result
If roots are real, they are x-intercepts of the parabola. If roots are complex, the parabola does not cross the x-axis.
Quick Example
Suppose the equation is x² - 3x + 2 = 0. Then:
- a = 1, b = -3, c = 2
- D = (-3)² - 4(1)(2) = 9 - 8 = 1
- x = (3 ± 1)/2 gives x = 2 and x = 1
You can test this instantly in the calculator by using the default values already loaded above.
Why This Tool Is Useful
- Fast way to solve ax² + bx + c = 0
- Great for algebra practice and exam review
- Shows discriminant and root type for better understanding
- Includes vertex and axis of symmetry for graph insights
Common Mistakes to Avoid
- Setting a = 0 (that makes the equation linear, not quadratic)
- Forgetting that -b means the opposite sign of b
- Dropping parentheses in b² - 4ac
- Ignoring complex solutions when the discriminant is negative
FAQ
Can this calculator handle decimals?
Yes. You can enter integers, fractions converted to decimals, or any real-number coefficient.
What if there are no real roots?
The calculator will return complex roots in the form p ± qi.
Does this method always work?
For all quadratic equations, yes. The quadratic formula is universal for degree-2 equations.