CCSS.Math.Content.HSN-CN.C.7
The standard
Solve quadratic equations with real coefficients that have complex solutions.
Common Core State Standards for Mathematics · The Complex Number System
What this standard means
Students need to solve quadratics that do not cross the x-axis. They should know when the discriminant is negative, the solutions are complex conjugates. They should use the quadratic formula, simplify square roots of negatives, and write answers in a plus bi form.
Mastery looks like choosing a method, checking the discriminant, and simplifying cleanly without stopping at a negative square root. Students often lose the i, make sign errors in the formula, or think “no real solution” means “no solution.” They also need practice seeing that real coefficients produce conjugate pairs.
Ways to teach it
- Hands-on activity: Use graphing calculators to graph three quadratics, then match each graph to its real or complex solutions from cards.
- Discussion prompt: Ask, “If a parabola never touches the x-axis, what kind of solutions can the equation still have?”
- Quick assessment: Give 2x^2 + 4x + 5 = 0 and ask students to solve and explain the negative discriminant.
- Real-world connection: Model projectile height with a quadratic and discuss why some time values are real while others are not physically possible.
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