The Quadratic Formula, Step by Step
Updated 2026-07-12
The quadratic formula solves any equation of the form ax^2 + bx + c = 0. The solutions are x = (-b plus or minus the square root of (b^2 - 4ac)), all divided by 2a. The part under the root, b^2 - 4ac, is the discriminant and tells you how many real solutions exist.
The quadratic formula always works, even when factoring fails. The key is reading off a, b, and c correctly (with their signs) and remembering the plus-or-minus, which is where the two solutions come from.
Reading off a, b, and c
Write the equation as ax^2 + bx + c = 0, then a is the number with x^2, b is the number with x, and c is the constant. Signs matter: in x^2 - 5x + 6, b is negative 5, not 5.
What the discriminant tells you
The discriminant b^2 - 4ac decides the outcome before you finish. Positive means two real solutions, zero means one repeated solution, negative means no real solutions (the roots are complex).
Worked example
Solve x^2 - 5x + 6 = 0.
- Identify a = 1, b = -5, c = 6.
- Discriminant: b^2 - 4ac = 25 - 24 = 1.
- Apply the formula: x = (5 plus or minus 1) / 2.
Answer: x = 3 and x = 2.
The common mistake
Wrong: Dropping the sign of b, using 5 instead of -5.
Fix: b keeps its sign from the equation. Since the term is -5x, b is negative 5, and -b becomes positive 5 in the formula.
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Frequently asked questions
What is the quadratic formula?
It solves ax^2 + bx + c = 0. The solutions are x equals negative b plus or minus the square root of (b squared minus 4ac), all over 2a.
What does the discriminant tell you?
The discriminant, b^2 - 4ac, tells you the number of real solutions: positive gives two, zero gives one repeated solution, and negative gives no real solutions.