Quadratic Formula Calculator with Steps
Enter coefficients a, b, and c to view discriminant Δ, roots, and step-by-step LaTeX solution.
The calculator above solves the quadratic equation ax² + bx + c = 0 and writes out each step: the coefficients, the discriminant, and the quadratic formula.
The quadratic formula
A quadratic equation in one variable has the standard form
where a, b, and c are known numbers, x is the unknown, and a ≠ 0. Its solutions are
The symbol ± means you compute the expression twice: once with the plus sign and once with the minus sign. If a coefficient is negative, keep the parentheses when you square it or multiply it.
The discriminant Δ
The value under the square root is called the discriminant:
- Δ > 0: two distinct real solutions.
- Δ = 0: one repeated real solution, also called a double root.
- Δ < 0: no real solution, because a negative number has no real square root.
Simplify √Δ before you divide by 2a. The exact radical form is the precise answer. A decimal rounded to two places is only a check.
Worked examples
Enter the same coefficients in the calculator at the top and compare the steps.
Example 1. Two distinct real solutions
Solve x² − 5x + 6 = 0. Here a = 1, b = −5, and c = 6.
Both values satisfy the original equation, and (x − 3)(x − 2) = x² − 5x + 6. The solution set is S = {3; 2}.
Example 2. A repeated root
Solve x² − 4x + 4 = 0. The coefficients are a = 1, b = −4, and c = 4.
The equation is the perfect square (x − 2)² = 0, so x = 2 is a double root.
Example 3. No real solution
Solve x² + 1 = 0. Here a = 1, b = 0, and c = 1.
Because Δ < 0, √Δ is not a real number. The equation has no real solution, and the solution set is S = ∅.
Frequently asked questions
What is the quadratic formula?
For ax^2 + bx + c = 0 with a not equal to 0, the solutions are x = (-b ± sqrt(b^2 – 4ac)) / (2a).
What does the discriminant tell you?
The discriminant is delta = b^2 – 4ac. A positive delta gives two distinct real solutions, zero gives one repeated real solution, and a negative delta gives no real solution.
What if a equals 0?
The equation is no longer quadratic. It becomes bx + c = 0. If b is not zero then x = -c/b. If b and c are both zero, every real number is a solution. If b is zero and c is not, there is no solution.
Should the square root be simplified?
Yes. Factor perfect squares out of the radicand and reduce the fraction. The exact radical form is the precise answer. A decimal is only a rounded check.
How many real solutions can a quadratic equation have?
At most two. When delta is positive there are two distinct real solutions, when delta is zero there is a repeated real solution, and when delta is negative there is no real solution.