Solve ax² + bx + c = 0 by entering the three coefficients. The calculator shows the roots, the discriminant, the vertex of the curve and the working.
Results are estimates for planning. Check the figures before you make a financial or health decision.
How the Quadratic equation solver works
The solver uses the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
The part under the root, b² − 4ac, is the discriminant:
- Above zero: two different real roots.
- Zero: one repeated real root, x = −b ÷ 2a.
- Below zero: no real roots. The calculator gives the two complex roots in the form p ± qi, where i is the square root of −1.
The vertex is the turning point of the parabola, at x = −b ÷ 2a. The y value is found by putting that x back into the equation. If a is positive the curve opens upwards and the vertex is a minimum; if a is negative it opens downwards and the vertex is a maximum. The tool also gives the axis of symmetry and the y-intercept, which equals c.
If a is 0 the equation is a straight line, bx + c = 0, and the single root is −c ÷ b. If a and b are both 0 there is no solution to show. Roots are rounded to six decimal places.
Worked example
Using the default values in the calculator above:
| Input | Value |
|---|---|
| a (the x² coefficient) | 1 |
| b (the x coefficient) | -3 |
| c (the constant) | -4 |
Roots of x² − 3x − 4 = 0: x = -1 or x = 4
| Discriminant (b² − 4ac) | 25 = (-3)² − 4 × 1 × (-4) |
|---|---|
| Type of roots | Two different real roots |
| Formula | x = (−b ± √D) ÷ 2a = (3 ± 5) ÷ 2 |
| Sum and product of roots | 3 and -4 |
| Vertex | (1.5, -6.25) |
| Axis of symmetry | x = 1.5 |
Tips and common mistakes
Write the equation in the form ax² + bx + c = 0 before you type the numbers, moving every term to one side. Keep the signs: x² − 3x − 4 = 0 has b = −3 and c = −4. If a term is missing, enter 0 for it.
Check an answer by substituting a root back into the equation; it should come to zero or very close. Roots with an i are not mistakes: they simply mean the curve never touches the x-axis. For a long calculation, the scientific calculator can help with the arithmetic.
Frequently asked questions
What does the discriminant tell me?
It shows how many real roots there are without solving the equation. Positive means two, zero means one repeated root, and negative means none, only complex roots.
What happens if a is zero?
The equation is no longer quadratic. It becomes a straight line, bx + c = 0, and the calculator solves it as x = −c ÷ b.
What is the vertex?
The vertex is the lowest point of an upward-opening parabola or the highest point of a downward one. It sits at x = −b ÷ 2a.