Montana Math MT.PLUS.ALG.POLY.4

MathGrades 9–12Polynomial Functions

The Standard

Solve quadratic equations that have complex solutions and understand why the solutions form a conjugate pair.

2026 Montana Mathematics Content Standards · Algebraic and Functional Reasoning

Teacher's field guide

What Students Need to Do: Students solve quadratic equations with real coefficients when the answers are complex numbers. They write both answers using i and explain why the imaginary parts have opposite signs.
What Mastery Looks Like: For x² + 4x + 13 = 0, a student finds −2 + 3i and −2 − 3i. They can check both by substitution and explain why conjugating a root still gives a solution when the coefficients are real.
Common Misconceptions: Some students write √−9 as −3 instead of 3i, or forget the ± when taking a square root. Others give only one root or think a complex number cannot solve an equation with real coefficients.
How to Assess It: Ask students to solve x² + 6x + 13 = 0. Have them write both roots and one sentence explaining why the second root must also work.

Lesson moves

Give pairs a printed complex-plane grid and two counters; have them plot −2 + 3i and −2 − 3i, then check both by substitution.
Ask: If 2 + 3i solves an equation with real coefficients, why does 2 − 3i also solve it? Write two sentences.
Make cards for x² + 4 = 0, x² + 2x + 5 = 0, and x² − 6x + 13 = 0; students match each to its two roots.
Compare revenue R(x) = 20x with cost C(x) = x² + 16x + 13; solve R = C and explain why the business never breaks even.

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