Enter coefficients for a quadratic equation in the form ax² + bx + c = 0, then click “Solve Quadratic.”
What this solve quadratic calculator does
A quadratic equation has the general form ax² + bx + c = 0, where a ≠ 0. This calculator helps you quickly find the roots (also called solutions or zeros) using the quadratic formula. It handles all major cases: two real roots, one repeated real root, and complex roots.
If you are studying algebra, preparing for exams, teaching math, or checking homework, this tool gives fast, reliable results and shows key details like the discriminant.
Quadratic formula refresher
The expression inside the square root, b² - 4ac, is called the discriminant. It determines the type of solution you get:
- Discriminant > 0: two distinct real roots
- Discriminant = 0: one repeated real root
- Discriminant < 0: two complex conjugate roots
How to use the calculator
Step 1: Enter coefficients
Type values for a, b, and c. These can be integers or decimals. Example: a = 2, b = 5, c = -3.
Step 2: Click “Solve Quadratic”
The calculator computes the discriminant and then applies the formula to return the roots. Results are displayed in a clean summary.
Step 3: Interpret the roots
Real roots can often be plotted as x-intercepts on a graph. Complex roots appear when the parabola does not cross the x-axis in the real coordinate plane.
Worked examples
Example 1: Two real roots
Equation: x² - 3x + 2 = 0
Discriminant: (-3)² - 4(1)(2) = 9 - 8 = 1
Roots: x = 1 and x = 2
Example 2: One repeated root
Equation: x² - 4x + 4 = 0
Discriminant: 16 - 16 = 0
Root: x = 2 (double root)
Example 3: Complex roots
Equation: x² + 2x + 5 = 0
Discriminant: 4 - 20 = -16
Roots: x = -1 + 2i and x = -1 - 2i
What if a = 0?
If a = 0, the equation is no longer quadratic. It becomes linear: bx + c = 0. This calculator detects that case and gives the linear solution when possible.
- If b ≠ 0: one linear solution x = -c / b
- If b = 0 and c = 0: infinitely many solutions
- If b = 0 and c ≠ 0: no solution
Common mistakes to avoid
- Forgetting that only the discriminant is under the square root.
- Dropping parentheses around negative numbers (especially b).
- Using 2a incorrectly in the denominator.
- Rounding too early and introducing avoidable error.
Final thoughts
A good quadratic equation solver should do more than output numbers—it should help you understand the structure of the equation. Use this tool to check your work, practice pattern recognition, and build intuition around discriminants, factoring, and graph behavior.
Whether you call it a quadratic roots calculator, polynomial solver, or quadratic formula calculator, the goal is the same: faster, clearer algebra.