Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 complex solutions of quadratics? Then this Algebra 2 lesson on Quadratics with Complex Solutions explains each idea step by step, with worked examples.
Quick answer: The discriminant D = b2 − 4ac of a quadratic ax2 + bx + c = 0 shows its solutions: D > 0 gives two real, D = 0 gives one real, D < 0 gives two complex conjugates. Apply the quadratic formula and write √D as i√(−D) to get exact answers.
Standards: N-CN.C.7, A-REI.B.4b · Level: Grade 11 Algebra 2
What does the discriminant tell you?
For a quadratic equation ax2 + bx + c = 0 with real coefficients, the discriminant is D = b2 − 4ac. Its sign decides what kind of solutions exist before any solving is done.
| Value of D | Solutions | Graph of y = ax2 + bx + c |
|---|---|---|
| D > 0 | Two real numbers | Crosses the x-axis twice |
| D = 0 | One repeated real number | Touches the x-axis once |
| D < 0 | Two conjugate complex numbers | Never meets the x-axis |
How do you solve a quadratic with complex solutions?
x = −b ± √(b2 − 4ac)2a
When D < 0, write √D = i√(−D) and simplify the radical.
- Write the equation as ax2 + bx + c = 0 and name a, b, c.
- Compute D = b2 − 4ac.
- If D < 0, replace √D with i√(−D) and simplify.
- Divide every part of the numerator by 2a and reduce.
Solve 2x2 − 4x + 5 = 0. Here D = 16 − 40 = −24 and √(−24) = 2i√6. Then x = 4 ± 2i√64 = 1 ± √62i.
Why do complex solutions come in pairs?
The ± in the formula produces two numbers with the same real part and opposite imaginary parts. Such numbers are complex conjugates, and they are mirror images across the real axis.
Do not stop at x = 4 ± √(−24)4. A root of a negative number must be rewritten with i, and any common factor must be reduced across the whole numerator and denominator.
Check any pair of solutions fast: they add to −b/a and multiply to c/a. For 2x2 − 4x + 5, the sum is 2 and the product is 52.
- D = b2 − 4ac decides the number and type of solutions.
- If D < 0, the parabola never crosses the x-axis and the solutions are non-real.
- Complex solutions of a real quadratic always appear as conjugates p + qi and p − qi.
- The quadratic formula gives exact solutions: x = −b ± √D2a.
- The two solutions sum to −b/a and multiply to c/a.
Quadratics with Complex Solutions: frequently asked questions
What does a negative discriminant mean?
A negative discriminant means the quadratic has no real solutions, so its graph does not cross the x-axis. The equation still has two solutions, and both are non-real complex numbers that are conjugates of each other.
Can a quadratic have just one complex solution?
Not if its coefficients are real. Non-real solutions come in conjugate pairs, so a quadratic with real coefficients has either two non-real solutions or real solutions.
How do you simplify the radical when D is negative?
Factor out −1 first: √(−24) = i√24. Then pull out perfect squares, so √24 = 2√6 and √(−24) = 2i√6.
Is completing the square or the quadratic formula better for complex roots?
Both always work. Completing the square is fast when the x-coefficient is even, as in x2 + 6x + 25 = 0, while the formula is safer when the coefficients are awkward.
How can you check complex solutions?
Use the sum and product shortcut: the solutions add to −b/a and multiply to c/a. You can also substitute one solution into the equation and confirm that the result is 0.
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