M151Q Precalculus · Unit 3: Quadratics, polynomials, triangles · on Exam 3
MFG 7.3 Complex numbers
Study notes for Complex numbers in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- $i = \sqrt{-1}$, $i^2 = -1$. Every complex number is $a + bi$. Square roots of negatives: $\sqrt{-20} = 2i\sqrt5$.
- Add and subtract componentwise; multiply like binomials and replace $i^2$ with $-1$; divide by multiplying top and bottom by the conjugate $a - bi$ (which makes the denominator $a^2 + b^2$).
- Quadratics with negative discriminant have two complex conjugate roots: $x^2 - 2x + 5 = 0 \Rightarrow x = 1 \pm 2i$. Every polynomial of degree $n$ has exactly $n$ zeros counting multiplicity and complex zeros.
Key formulas
Arithmetic
$$(a+bi)(c+di) = (ac - bd) + (ad + bc)i,\qquad \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2}$$
Powers of i
$$i^2 = -1,\quad i^3 = -i,\quad i^4 = 1$$
Worked example
Compute $(3 - 2i)(1 + 4i)$ and $\dfrac{2 + i}{3 - i}$, then solve $x^2 + 4x + 13 = 0$.
Show the solution
$(3-2i)(1+4i) = 3 + 12i - 2i - 8i^2 = 11 + 10i$. $\dfrac{(2+i)(3+i)}{9+1} = \dfrac{5 + 5i}{10} = \dfrac12 + \dfrac12 i$. $x = \dfrac{-4\pm\sqrt{16-52}}{2} = -2 \pm 3i$.
Common mistakes
- Writing $i^2 = 1$.
- $\sqrt{-4}\sqrt{-9} = \sqrt{36} = 6$ (wrong; it is $(2i)(3i) = -6$).
Exam tipTreat i like a variable when multiplying, then clean up i² = −1 at the end.
Keep going in Mathub
Mathub is free for Montana State students: an endless quizzer with worked explanations, one-question lessons, flashcards, exam prep checklists, the class calendar with every deadline, a grade calculator and a study board for M151Q.