Enter the coefficients a, b and c from the equation ax² + bx + c = 0.
The calculator automatically works out the discriminant and the solutions as you type.
Read the explanation below to understand what the discriminant tells you.
How the quadratic equation is solved
Every quadratic equation of the form ax² + bx + c = 0 can be solved with the quadratic formula:
x = (−b ± √(b² − 4ac)) ÷ (2a)
The expression under the square root, b² − 4ac, is called the discriminant and determines how many real solutions the equation has:
Positive discriminant — the equation has two real solutions.
Discriminant equal to zero — the equation has one solution (a double root).
Negative discriminant — the equation has no real solutions (the solutions are complex numbers).
Note: a cannot be 0 — that would remove the x² term, and the equation would no longer be a quadratic equation.
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Frequently asked questions
What does the discriminant tell me?
The discriminant (b² − 4ac) tells you how many real solutions the equation has: positive gives two solutions, zero gives one (a double root), and negative gives no real solutions.
What happens if a is 0?
Then the equation is no longer a quadratic equation, since the x² term disappears. The calculator shows an error message in this case — use a regular linear equation instead.
Are there solutions when the discriminant is negative?
Not real solutions. The equation then has two complex solutions, involving the square root of a negative number (imaginary numbers). This calculator only shows real solutions.
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