OnlineCalculatorLinear equations
An equation of the form ax + b = 0 has exactly one solution, found by isolating x: subtract b from both sides, then divide by a. The only case with no solution is a = 0 with b non-zero, which reduces to a false statement.
Quadratic equations and the discriminant
For ax² + bx + c = 0, the quadratic formula gives x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant and it determines the nature of the roots before you calculate them:
| Discriminant | Roots |
|---|---|
| Positive | Two distinct real roots |
| Zero | One repeated real root |
| Negative | Two complex conjugate roots |
The vertex
A quadratic graphs as a parabola, and its turning point sits at x = −b / 2a. Substituting that back into the equation gives the corresponding y value. If a is positive the parabola opens upward and the vertex is a minimum; if negative, it opens downward and the vertex is a maximum.
Checking your answer
Substitute each root back into the original equation. A correct root produces zero. This catches arithmetic slips faster than re-deriving the whole solution.