Quick start
- Type the number in front of x² as a, the number in front of x as b, and the constant as c. For 2x² + 5x − 3 = 0, that is 2, 5 and −3.
- Check the discriminant line. It decides what kind of roots you will get.
- Read the roots, the factored form, the vertex and the sum and product of the roots.
The formula behind it
Every quadratic can be solved with x = (−b ± √(b² − 4ac)) ÷ 2a. Some countries call this the ABC formula, after the three letters it uses. The ± gives you two answers: one where you add the square root and one where you subtract it.
Reading the discriminant
| b² − 4ac | What you get | Graph |
|---|---|---|
| Positive | Two different real roots | Crosses the x-axis twice |
| Zero | One repeated root | Touches the x-axis once |
| Negative | Two complex roots | Never meets the x-axis |
Example: 2x² + 5x − 3 = 0
Here a = 2, b = 5 and c = −3. The discriminant is 25 + 24 = 49, and √49 = 7. So x = (−5 + 7) ÷ 4 = 0.5 or x = (−5 − 7) ÷ 4 = −3. In factored form, 2(x − 0.5)(x + 3) = 0, which is the same as (2x − 1)(x + 3) = 0.
Where the vertex comes from
The graph of a quadratic is a parabola, and its turning point sits at x = −b ÷ 2a, exactly halfway between the two roots. Plug that x back in to get the y-value. If a is positive the parabola opens upward and the vertex is the lowest point; if a is negative it is the highest point.
Common questions
Why does the solver say a cannot be 0?
Without the x² term the equation is linear, not quadratic. Solve it with the Linear Equation Solver instead.
What do complex roots look like?
They come in pairs written p ± qi, where i is the square root of −1. They appear whenever b² − 4ac is negative.
Is the ABC formula the same as the quadratic formula?
Yes. It is just another name for x = (−b ± √(b² − 4ac)) ÷ 2a.
How can I check my roots?
Add them: the total should equal −b/a. Multiply them: the product should equal c/a. The solver prints both checks.