Quadratic Equation Solver
Roots, discriminant, and vertex for any ax squared plus bx plus c.
What it does
Quadratic equations curve into parabolas, and their roots mark where the curve crosses the x axis. The quadratic formula answers every case at once by folding the coefficients into one expression with a square root at its heart.
The tool above takes a, b, and c. It returns the roots in real or complex form, the discriminant, and the vertex with the opening direction.
The discriminant earns its name by deciding the story upfront. Positive values give two crossings, zero grazes the axis once, and negative values miss it entirely, leaving a conjugate complex pair.
Completing the square hides behind the formula and explains where it comes from. Shifting the parabola to its vertex turns the equation into a perfect square plus a constant, and the formula drops out.
How to use
- Move everything to one side so the form reads ax squared plus bx plus c equals zero.
- Enter a, b, and c with their signs.
- Read roots first, then discriminant and vertex.
- For word problems, discard roots outside the allowed domain.
Formula
- a, b, c
- Coefficients of x squared, x, and the constant.
- D
- Discriminant deciding root count and type.
- ±
- Both plus and minus branches give the two roots.
Dividing by 2a fails at a equals zero, which is why the tool treats that as linear. Complex roots always pair as conjugates with matching real parts.
Worked example
Example: x squared minus 5x plus 6
Coefficients a 1, b -5, c 6.
- Discriminant: (-5)² - 4 x 1 x 6 = 25 - 24 = 1.
- Root one: (5 - 1) / 2 = 2.
- Root two: (5 + 1) / 2 = 3.
- Answer: x = 2 and x = 3.
- Vertex: x = 5/2 = 2.5, y = 6 - 25/4 = -0.25. Point (2.5, -0.25), opening upward.
Reference table
| D value | Roots | Graph |
|---|---|---|
| D > 0 | Two real roots | Crosses twice |
| D = 0 | One double root | Touches once |
| D < 0 | Complex pair | Never touches |
Limits
- Exact to floating precision. Huge coefficients lose digits.
- Real world domains can reject valid math roots.
- Linear case handled separately, not by the formula.
- Graphing still helps for inequalities around roots.
Common mistakes
- Dropping signs on b and c.
- Forgetting the 2a denominator under both terms.
- Taking the root of a negative discriminant as real.
- Stopping at one root when two exist.
- Reading vertex x as a root.
Try it yourself
Quick checks. Answers sit tucked under each question.
Roots of x² - 4?
D = 16. x = ±2.
Roots of x² + 2x + 5?
D = 4 - 20 = -16. x = -1 ± 2i.
Questions
What is the discriminant?
b² - 4ac. Its sign tells whether roots are real double, real split, or complex.
What are complex roots?
Pairs with imaginary parts, like -1 ± 2i, when the parabola misses the axis.
How do I factor with roots?
a(x - r1)(x - r2). Roots 2 and 3 give (x - 2)(x - 3).
What if a is zero?
The equation is linear bx + c = 0 with root -c/b. The tool switches automatically.
Where is the vertex used?
Max and min problems: profit peaks, projectile tops, area extremes.
Can coefficients be decimals?
Yes. The formula works for any real numbers, fractions included.