Quadratics with complex solutionsBoth kinds
Name: Date:
Solve each equation. Every answer is a complex number.
1x² + 6x + 10 = 0
2x² + 9 = 0
3x² + 64 = 0
4x² − 6x + 10 = 0
5x² + 25 = 0
6x² + 8x + 17 = 0
7x² + 10x + 41 = 0
8x² − 4x + 68 = 0
Quadratics with complex solutions worksheets
When the discriminant is negative the parabola misses the axis and the roots leave the real numbers. These free printable worksheets take that case on its own, from x² + 9 = 0 to the full quadratic formula.
If they get stuck
Quadratics with complex solutions
The answer key prints on a separate page.
What's on this sheet
The discriminant comes first on every key, and out loud. b² − 4ac being negative is the whole reason the answer changes kind, and a solution that arrives at ± 3i without showing it has skipped the only interesting step. Counting real roots is a page of its own elsewhere; this is what happens when the count is zero.
Why this sheet works
√(−36) = 6i is the move students undo wrongly — writing −6, or √6 i — so it gets its own line rather than being folded into the formula’s arithmetic.
How this one works
Solve x² + 6x + 10 = 0
b² − 4ac = −4
x = (−6 ± √−4) ÷ 2
x = (−6 ± 2i) ÷ 2
x = −3 ± i
- Work out the discriminant first: b² − 4ac = 6² − 4(10) = 36 − 40 = −4. It is negative, so there are no real roots.
- Put it in the formula: x = (−6 ± √−4) ÷ 2.
- √−4 = 2i, because √(−1) is i. So x = (−6 ± 2i) ÷ 2.
- Divide both parts by 2: x = −3 ± i.