A quadratic equation contains a squared variable and can be written as ax² + bx + c = 0, where a is not zero. The quadratic formula finds the solutions, also called roots, even when an equation does not factor easily.
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / (2a)
The symbol ± means that you calculate two possibilities: one using addition and one using subtraction. Depending on the equation, these may be two distinct real roots, one repeated real root, or two complex roots.
How to use a quadratic formula calculator
- Open the Quadratic Formula Calculator.
- Rewrite the equation so the right-hand side is zero.
- Identify coefficients a, b, and c, including negative signs.
- Enter the coefficients and calculate.
- Check the roots by substituting them into the original equation.
Worked example: two real roots
Solve x² − 5x + 6 = 0. Here a = 1, b = −5, and c = 6. The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1. Thus x = (5 ± 1)/2, giving x = 3 and x = 2. Substitution confirms that both values satisfy the equation.
What is the discriminant?
The expression D = b² − 4ac is called the discriminant. It indicates the type of roots:
| Discriminant | Meaning |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | One repeated real root |
| D < 0 | Two non-real complex conjugate roots |
Example with one repeated root
For x² − 4x + 4 = 0, the discriminant is 16 − 16 = 0. The formula gives x = 2 for both signs. This is a repeated root.
Example with complex roots
For x² + 1 = 0, a = 1, b = 0, and c = 1. The discriminant is −4. The solutions are x = i and x = −i, where i² = −1. Some online calculators display complex solutions differently or may only support real roots.
Common mistakes
- Entering b without its negative sign.
- Forgetting parentheses around the entire numerator.
- Dividing only the square-root term by 2a.
- Trying to apply the formula when a = 0; that equation is not quadratic.
- Assuming a negative discriminant means the equation has no solutions at all; it has no real solutions but does have complex solutions.
Frequently asked questions
Does the quadratic formula work for every quadratic equation?
Yes, over the complex numbers, provided a is not zero.
What if b or c is zero?
The formula still works. Substitute zero for the missing coefficient.
Why do I sometimes get only one answer?
When the discriminant is zero, the two expressions produce the same repeated root.
Can I solve quadratics by factoring instead?
Yes. Factoring is often faster when simple factors exist, while the quadratic formula is a general method.
Check your coefficients and solutions using the Quadratic Formula Calculator.
Moiz Ahmad is a digital marketer and SEO expert based in Dera Ismail Khan, Pakistan. With over 6 years of experience in SEO, WordPress development, and digital strategy, he has helped 30+ businesses rank on Google’s first page. He is the founder of DigiTechPak and Moiz Blogger. Connect: LinkedIn | Twitter | Moiz Blogger