Transparent method
The formula
Worked example
Example inputs
How to interpret the result
The discriminant, the piece under the square root, decides the whole shape of the answer before the rest of the formula even runs. A positive value gives two distinct real roots, zero gives exactly one repeated root, and a negative value means the parabola never touches the x-axis at all: the two roots come out as a complex conjugate pair instead of two real numbers.
Before you rely on it
What to check
Read the discriminant's sign before trusting the numeric roots. A negative discriminant means the real-number roots this calculator shows aren't what's being asked for: the equation has none.
The common error
Where people go wrong with quadratic formula calculator
Expecting a negative discriminant to mean an input error. It's a legitimate mathematical outcome, not a bug, and it's telling you something specific about the parabola's position relative to the x-axis.
Sensitivity evidence
How a (x² coefficient) changes the root 1 (x₁)
| a (x² coefficient) | Root 1 (x₁) | Root 2 (x₂) |
|---|---|---|
| 0.8 | 2.9 | 0.867 |
| 0.9 | 2.4 | 0.921 |
| 1worked example | 2 | 1 |
| 1.1 | 1.6 | 1.2 |
| 1.2 | : | : |
Every input, tested
Which input moves the root 1 (x₁) most
| Input | Tested from | To | Root 1 (x₁) at each end | Swing |
|---|---|---|---|---|
| a (x² coefficient) | 0.9 | 1.1 | 2.4 to 1.6 | 0.845 (42%) |
| c (constant) | 1.8 | 2.2 | 2.2 to 1.7 | 0.447 (22%) |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| a (x² coefficient) | 1 |
| b (x coefficient) | -3 |
| c (constant) | 2 |
| Root 1 (x₁) | 2 |
| Root 2 (x₂) | 1 |
Where the inputs come from
Write the equation in the form ax² + bx + c = 0 before entering anything, moving every term to one side, so that a is the number multiplying x², b multiplies x and c is the constant. If the equation is 2x² = 8x − 6, subtract to get 2x² − 8x + 6 = 0, giving a = 2, b = −8 and c = 6. The coefficient a cannot be zero, since that makes the equation linear. Keep the signs: a missing minus sign is the most common error.
When the result stops applying
When the discriminant is negative the roots are complex numbers, and a calculator restricted to real answers will report none. Very large or very small coefficients can cause rounding errors, and subtracting nearly equal numbers loses precision. For word problems, a root may be mathematically valid but meaningless in context, such as a negative length or time. Factoring is faster when the roots are whole numbers.
What to compare it with
Rewrite the equation to find the vertex on the completing the square calculator, and check simple cases with the exponent and root calculators. The distance and slope page handles lines, and the exponential growth calculator covers another common model. For graph shapes, the vertex and the roots together describe the parabola.
People also ask
Frequently asked questions
What does the discriminant tell me about a quadratic equation?
It shows how many real solutions exist: two if it is positive, one repeated root if zero, and none, only complex roots, if it is negative.
Can a quadratic equation have only one solution?
Yes. When the discriminant is zero, both roots are the same number and the parabola just touches the x-axis at its vertex.
Why must the coefficient a not be zero?
If a is zero the x-squared term disappears and the equation is linear, which has one solution and is not solved by the quadratic formula.