How to use it
- Enter the coefficient a (of x²).
- Enter b (of x) and the constant c.
- Read the roots and the discriminant.
- Check the steps, the vertex and the factored form.
The quadratic formula
Every equation of the form ax² + bx + c = 0 (with a ≠ 0) can be solved with the quadratic formula:
x = (−b ± √(b² − 4ac)) / 2a
The expression under the square root, D = b² − 4ac, is the discriminant. It tells you what kind of solutions to expect:
| Discriminant | Solutions |
|---|---|
| D > 0 | Two different real roots |
| D = 0 | One repeated real root |
| D < 0 | Two complex conjugate roots, no real solution |
If you enter a = 0, the equation is linear (bx + c = 0) and the calculator solves that instead. All the work happens in your browser.
Worked example
Solve x² − 5x + 6 = 0, so a = 1, b = −5, c = 6.
- D = (−5)² − 4 · 1 · 6 = 25 − 24 = 1
- x = (5 ± √1) / 2 = (5 ± 1) / 2
- x₁ = 3, x₂ = 2
Factored form: (x − 3)(x − 2) = 0. Vertex: (2.5, −0.25); since a > 0, the parabola opens upward.
Repeated root: x² − 6x + 9 = 0 has D = 0 and the single root x = 3.
Complex roots: 2x² + 4x + 5 = 0 has D = 16 − 40 = −24, giving x = −1 ± 1.2247i.
Word problem: when does the ball land?
A ball is thrown upward from 4 feet with a speed of 48 feet per second. Its height in feet after t seconds is h = −16t² + 48t + 4. When does it hit the ground (h = 0)?
Enter a = −16, b = 48, c = 4:
- D = 48² − 4 · (−16) · 4 = 2,304 + 256 = 2,560
- t = (−48 ± √2,560) / (−32), giving t ≈ −0.081 and t ≈ 3.081
Time cannot be negative, so the ball lands after about 3.08 seconds. The vertex, at t = 1.5 s and h = 40 ft, is the highest point.
The vertex and the graph
The graph of y = ax² + bx + c is a parabola. Its vertex, the highest or lowest point, is at
x = −b / 2a, y = c − b² / 4a
If a is positive, the parabola opens upward and the vertex is a minimum; if a is negative, it opens downward and the vertex is a maximum. The real roots are where the parabola crosses the x-axis. To see this, plot the equation with the graphing calculator.
Other ways to solve quadratics
- Factoring is fastest when the roots are small whole numbers: x² − 5x + 6 = (x − 2)(x − 3).
- Completing the square rewrites the equation as a(x − h)² + k = 0 and is how the quadratic formula is derived.
- Graphing gives approximate roots and a feel for the shape.
The quadratic formula always works, which is why it is worth memorizing. For square roots that appear in exact answers, the square root calculator can simplify radicals such as √24 = 2√6.
Common mistakes
- Sign errors with b. If b is negative, −b is positive. For x² − 5x + 6, −b = 5.
- Dividing only part of the numerator by 2a. The whole expression −b ± √D goes over 2a.
- Forgetting to set the equation to zero. Rewrite x² = 5x − 6 as x² − 5x + 6 = 0 before reading off a, b and c.
- Squaring a negative b. (−5)² = 25, not −25.
- Discarding valid roots. Both roots solve the equation; only the context (such as time being positive) tells you which one to keep.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / 2a, which gives the solutions of ax² + bx + c = 0.
What does the discriminant tell me?
If b² − 4ac is positive there are two real roots, if it is zero there is one repeated root, and if it is negative the roots are complex.
What if a = 0?
The equation is linear, bx + c = 0, and the calculator solves it as such.
Does the calculator give complex roots?
Yes. When the discriminant is negative, it shows the roots in the form p ± qi.
How do I find the vertex of a parabola?
The x-coordinate is −b / 2a; plug it into the equation for the y-coordinate. The calculator shows both.
Last updated: November 12, 2026