Quadratic Formula Calculator · Practical guide
How to Find the Roots of a Quadratic Equation
Write the equation as ax² + bx + c = 0, then plug a, b, and c into x = (−b ± √(b² − 4ac)) / 2a.
At a glance
Quick answer
To find the roots of a quadratic equation, put it in standard form ax² + bx + c = 0 and apply the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. For x² − 5x + 6 = 0, the roots are x = 2 and x = 3. Enter your coefficients above to get the roots, discriminant, and step-by-step work instantly.
Interactive tool
Quadratic Formula Calculator
Enter your own values, then compare the result with the examples below.
Solution
Step-by-Step:
Step-by-Step Example
For x² − 5x + 6 = 0, read off a = 1, b = −5, c = 6. The discriminant is D = (−5)² − 4(1)(6) = 25 − 24 = 1. Since D > 0, there are two real roots: x = (5 ± √1) / 2 = (5 ± 1) / 2, giving x = 3 and x = 2. Both check: 2² − 10 + 6 = 0 and 3² − 15 + 6 = 0.
Three Root Scenarios
| Equation | Discriminant | Roots |
|---|---|---|
| x² − 5x + 6 = 0 | D = 1 | x = 2, x = 3 |
| x² − 6x + 9 = 0 | D = 0 | x = 3 (repeated) |
| x² + 2x + 5 = 0 | D = −16 | x = −1 ± 2i |
| x² − 2 = 0 | D = 8 | x = ±√2 ≈ ±1.4142 |
Getting the Coefficients Right
The most common mistake is reading b or c with the wrong sign. Rewrite everything on one side first: x² = 5x − 6 becomes x² − 5x + 6 = 0, so b = −5, not 5. If a term is missing, its coefficient is 0 — for 3x² − 12 = 0, use a = 3, b = 0, c = −12, giving x = ±2.
When Factoring Is Faster
If a = 1 and you can spot two numbers that multiply to c and add to b, factor instead: x² − 5x + 6 = (x − 2)(x − 3). Reach for the formula — or the calculator above — when the discriminant is not a perfect square or the coefficients are decimals. The formula never fails; factoring only works when the roots are rational.
Frequently Asked Questions
What if the equation is not in standard form?
Move every term to one side so it reads ax² + bx + c = 0 first. For example, 2x² = 3x + 5 becomes 2x² − 3x − 5 = 0, with a = 2, b = −3, c = −5.
Can the roots be the same number?
Yes. When the discriminant is zero, both branches of the ± give the same value, called a repeated or double root. For x² − 6x + 9 = 0, that root is x = 3.
How do I check my answer?
Substitute each root back into the original equation — it should evaluate to zero. You can also verify that the roots add to −b/a and multiply to c/a.
What if a equals zero?
Then it is not a quadratic — the x² term vanishes and you have a linear equation, solvable by simple rearrangement. The calculator will flag this case.