The Quadratic Formula over the Complex Numbers
The quadratic formula, , solves every quadratic — even ones that do not factor.
The piece under the root, the discriminant , previews the answer: positive gives two real solutions, zero gives one, and negative gives a complex pair.
Applying the formula
Identify , , and from standard form, then substitute into the formula and simplify. Take care with signs, especially a negative .
The produces the two solutions. Reduce the fraction and the radical at the end.
The discriminant
The discriminant decides the nature of the roots. Positive means two distinct real solutions; zero means one repeated real solution.
Negative means the root is imaginary, so the two solutions are complex conjugates like . Checking the discriminant first tells you what kind of answer to expect.
Worked examples
Example 1: two real solutions
Solve with the quadratic formula.
Answer: or
Example 2: a complex pair
What does a discriminant of tell you?
Answer: Two complex-conjugate solutions
Example 3: a discriminant of zero
Solve with the quadratic formula.
Answer: , one repeated real solution
Try one yourself
Common questions
When should I use the quadratic formula?
Any time a quadratic is hard to factor, or when you need exact solutions. It works on every quadratic in standard form.
What does the discriminant tell me?
reveals the solution type: positive → two real, zero → one real (repeated), negative → two complex.
How do complex solutions appear?
When the discriminant is negative, the square root produces , giving a conjugate pair like .
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