Quadratic Formula Calculator
Enter coefficients for an equation in standard form: ax2 + bx + c = 0.
How to Solve Quadratic Equations Quickly
A quadratic equation is any equation that can be written as ax2 + bx + c = 0, where a ≠ 0. The calculator above helps you solve for the roots (also called solutions or zeros) in seconds, including cases with two real solutions, one repeated solution, or two complex solutions.
The core formula
The expression under the square root, b² - 4ac, is called the discriminant. It tells you what kind of roots to expect before you even finish the calculation.
What the Discriminant Means
- If b² - 4ac > 0: two distinct real roots.
- If b² - 4ac = 0: one real repeated root (sometimes called a double root).
- If b² - 4ac < 0: two complex conjugate roots.
Step-by-Step Example
Suppose we want to solve:
- a = 2, b = 5, c = -3
- Discriminant: b² - 4ac = 25 - 4(2)(-3) = 25 + 24 = 49
- Square root: √49 = 7
- Roots: x = (-5 ± 7) / 4
- So the answers are x = 1/2 and x = -3
Why Use a Quadratic Calculator?
While solving by hand is important for learning, a calculator is useful for checking homework, validating algebra steps, and speeding up technical work in physics, engineering, statistics, and finance models.
Benefits
- Instant answers with less arithmetic error.
- Automatically identifies real vs. complex roots.
- Great for test prep and classroom practice.
- Handles decimal coefficients and negative values.
Common Mistakes to Avoid
- Forgetting that a cannot be zero for a quadratic.
- Sign errors when substituting negative b or c.
- Not using parentheses in calculations such as -b ± √(...).
- Stopping early when the discriminant is negative (complex roots are still valid solutions).
FAQ
Can this calculator solve linear equations too?
Yes. If you enter a = 0, the tool will treat it as a linear equation bx + c = 0 and solve it when possible.
What if all coefficients are zero?
Then every real number satisfies the equation. The calculator reports this special case.
Do I need integers only?
No. You can use integers, fractions converted to decimals, and negative numbers.