Quadratic equation solver
Enter the three coefficients of ax² + bx + c = 0 and see both solutions, whether they are real or complex, and exactly how the quadratic formula produced them.
How the quadratic equation solver works
Move every term to one side so the equation equals zero, then read off a, b and c. The calculator first works out the discriminant, b² - 4ac, which tells you what kind of answers to expect before you do any square roots: positive means two real solutions, zero means one repeated solution, and negative means no real solutions (two complex ones).
It also reports the vertex, the turning point of the parabola, which is handy for graphing and for problems that ask for a maximum or minimum.
The formula behind it
x = (-b ± √(b² - 4ac)) ÷ 2a
Vertex: x = -b ÷ 2a, and y is found by putting that x back into the equation.
A worked example
Solve x² - 5x + 6 = 0, so a = 1, b = -5, c = 6. The discriminant is 25 - 24 = 1. Then x = (5 ± 1) ÷ 2, giving x = 3 or x = 2. Check: 3² - 15 + 6 = 0. The vertex sits at (2.5, -0.25).
Questions people ask
What if a is zero?
Then there is no x² term and the equation is a straight line, not a quadratic. The calculator tells you the single answer in that case.
What does a negative discriminant mean?
The parabola never crosses the x-axis, so there are no real solutions. In school you usually write "no real roots"; at higher levels the answers are complex numbers.
Can I factorise instead?
Often, yes. If you can find two numbers that multiply to c and add to b, factorising is quicker. The formula always works, even when factorising does not.
What are the roots?
The roots are just the solutions: the x-values where the expression equals zero.