Quadratic Formula Calculator
Enter a, b and c for any equation of the form ax² + bx + c = 0. You get the roots in exact and decimal form, the discriminant, the vertex, a graph, and every step of the working.
The quadratic formula
Every quadratic equation ax2 + bx + c = 0, with a ≠ 0, is solved by
x = −b ± √b2 − 4ac2a
The ± means there are usually two answers: one using + and one using −. The formula works for every quadratic, including ones that cannot be factored and ones whose roots are complex numbers.
How to use the quadratic formula
- Rearrange the equation so everything is on one side and it equals zero: ax2 + bx + c = 0.
- Read off a, b and c, including their signs. In 2x2 − 6x + 3 = 0, a = 2, b = −6 and c = 3.
- Work out the discriminant, b2 − 4ac. Here it is (−6)2 − 4(2)(3) = 36 − 24 = 12.
- Substitute into the formula: x = (6 ± √12) ÷ 4.
- Simplify. Since √12 = 2√3, x = (6 ± 2√3) ÷ 4 = (3 ± √3) ÷ 2, which is about 2.366 or 0.634.
The calculator above shows exactly these steps for any equation you enter.
What the discriminant tells you
The discriminant D = b2 − 4ac is the part under the square root. Its sign tells you what kind of roots to expect before you finish solving.
| Discriminant | Roots | Graph | Example |
|---|---|---|---|
| D > 0 and a perfect square | Two different rational roots | Crosses the x-axis twice | x2 − 5x + 6 = 0 gives 2 and 3 |
| D > 0, not a perfect square | Two different irrational roots | Crosses the x-axis twice | x2 − 2x − 1 = 0 gives 1 ± √2 |
| D = 0 | One repeated real root | Touches the x-axis at the vertex | 4x2 − 12x + 9 = 0 gives 3/2 |
| D < 0 | Two complex conjugate roots | Never touches the x-axis | x2 + 2x + 5 = 0 gives −1 ± 2i |
Where the formula comes from
The quadratic formula comes from completing the square. Divide ax2 + bx + c = 0 by a, move c/a to the right side, and add (b/2a)2 to both sides. The left side becomes the perfect square (x + b/2a)2, and the right side becomes (b2 − 4ac)/4a2. Taking the square root of both sides and subtracting b/2a gives the formula.
Vertex, axis of symmetry and factored form
The graph of y = ax2 + bx + c is a parabola. Its vertex is at x = −b/2a, which is also the axis of symmetry, the vertical line that splits the parabola into mirror halves. The roots always sit the same distance on either side of it. If a is positive the parabola opens upward and the vertex is the minimum; if a is negative it opens downward and the vertex is the maximum.
When the roots r1 and r2 are rational, the quadratic factors as a(x − r1)(x − r2). The calculator shows this factored form whenever it exists. The roots also satisfy two handy checks: r1 + r2 = −b/a and r1r2 = c/a.
Worked example: a projectile
A ball is thrown upward from a 48-foot platform at 32 feet per second. Its height after t seconds is h = −16t2 + 32t + 48. When does it hit the ground?
Set h = 0, so a = −16, b = 32 and c = 48. The discriminant is 322 − 4(−16)(48) = 1024 + 3072 = 4096, and √4096 = 64. The formula gives t = (−32 ± 64) ÷ (−32), so t = −1 or t = 3. Time cannot be negative, so the ball lands after 3 seconds. The vertex, at t = 1, tells you it reaches its highest point, 64 feet, after one second. Try it with the last example button above.
Common mistakes
- Forgetting that b includes its sign. In x2 − 5x + 6, b is −5, so −b is +5.
- Squaring a negative b wrong. (−5)2 is 25, not −25. Put negative numbers in parentheses before squaring.
- Dividing only part of the top by 2a. The whole numerator, −b ± √D, is divided by 2a.
- Not setting the equation to zero first. For 3x2 = 5x + 2, subtract to get 3x2 − 5x − 2 = 0 before reading off a, b and c.
Frequently asked questions
What if b or c is missing?
Enter 0 for it. For x2 − 9 = 0, use a = 1, b = 0 and c = −9 to get x = ±3.
Can the calculator handle fractions?
Yes. Type fractions like 1/2 or −3/4. The steps start by multiplying through to clear the fractions, and the answers stay exact.
What does it mean when the answer has an i in it?
The discriminant is negative, so there are no real solutions. The roots are complex numbers, where i = √−1, and the parabola never crosses the x-axis.
What happens if a = 0?
The equation is no longer quadratic. It becomes the linear equation bx + c = 0, and the calculator solves that instead.
Can I share a problem with someone?
Yes. After you press Solve, the page address includes your a, b and c values, so copying the link opens the same problem.
Need to solve cubic equations or systems of equations? Use the equation mode in the fx-991CW style calculator.
Last updated September 27, 2026.