Quadratic Equation Solver
Solve ax² + bx + c = 0 for any coefficients using the quadratic formula — real roots, a repeated root, or complex roots, plus the discriminant and vertex.
Roots
x = 1, x = 2
Two distinct real roots
Discriminant (b² − 4ac)
1
Vertex
(1.5, −0.25)
Uses the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. When the discriminant is negative, the roots are complex — shown in a ± bi form — rather than treated as "no solution." If a = 0, the equation isn't actually quadratic; it's solved as a linear equation instead (or flagged if b is also 0).
About this tool
How to use it
- Enter the coefficients a, b, and c from your equation in the form ax² + bx + c = 0.
- Press Solve Equation, or just keep typing — the result updates live.
- Read the roots and whether they're real or complex, plus the discriminant and the vertex of the parabola.
Why it matters
The quadratic formula solves any equation of the form ax² + bx + c = 0, but working through the arithmetic by hand is error-prone, especially with negative coefficients or a negative discriminant. Seeing the discriminant's sign alongside the roots also makes it clear *why* an equation has two real roots, one repeated root, or no real roots at all, rather than just producing a final answer.
When to use it
Useful for checking homework or exam practice, for quickly solving an equation that comes up while working through a larger problem, or for exploring how changing a coefficient changes the number and nature of the roots.
Good to know
Uses the standard quadratic formula, x = (−b ± √(b² − 4ac)) / 2a. When the discriminant (b² − 4ac) is negative, there are no real solutions — the roots are complex numbers, shown in a ± bi form rather than reported as 'no solution', since a quadratic always has exactly two roots when complex numbers are allowed. If a = 0, the equation isn't actually quadratic; it's solved as the linear equation bx + c = 0 instead, and if both a and b are 0 the tool reports either 'no solution' or 'every x is a solution' depending on c.
Common questions
What does it mean if the discriminant is negative?
A negative discriminant means the equation has no real number solutions — the parabola never crosses the x-axis. It still has two solutions among the complex numbers, shown in a ± bi form, since taking the square root of a negative number is defined once imaginary numbers are allowed.
What if a = 0?
With a = 0 the equation isn't quadratic anymore — it's linear (bx + c = 0), so it's solved that way instead, with a single root, unless b is also 0, in which case there's either no solution or every value of x works, depending on c.
What is the discriminant used for?
The discriminant, b² − 4ac, tells you the nature of the roots without fully solving the equation: positive means two distinct real roots, zero means one repeated real root, and negative means two complex (non-real) roots.
What is the vertex shown in the results?
The vertex is the highest or lowest point of the parabola y = ax² + bx + c, at x = −b/2a. It's shown alongside the roots because it's often needed for the same graphing or optimization problems that ask you to solve the equation.
Why are the two roots shown in a particular order?
When there are two distinct real roots, they're shown smallest first for consistency — the quadratic formula's ± naturally produces them in whichever order the arithmetic happens to fall, so they're sorted before display rather than left in that arbitrary order.