Complex Numbers and the Quadratic Formula: Chapter Review

Vocabulary and notation

The imaginary unit $i$

The number defined by $i^2 = -1$, equivalently $i = \sqrt{-1}$.

Pure imaginary number

A real multiple of $i$, such as $3i$ or $-\tfrac{1}{2}i$.

Complex number, standard form $a + bi$

A real part $a$ plus an imaginary part $b$. Every answer should end in this form; a quotient with $i$ still in the denominator is unfinished.

Real and imaginary parts

For $z = a + bi$: $\operatorname{Re}(z) = a$ and $\operatorname{Im}(z) = b$, a real number, not $bi$.

Equality of complex numbers

$a + bi = c + di$ exactly when $a = c$ and $b = d$: the parts match separately.

Complex conjugate $\overline{z}$

For $z = a + bi$, the conjugate is $a - bi$: only the imaginary part flips sign.

Completed-square form $(x - h)^2 + k$

A quadratic rewritten as a perfect square plus a constant, solvable by one square root.

Standard form $ax^2 + bx + c = 0$

A quadratic set equal to zero, with $a \neq 0$. Read $a$, $b$, $c$, with their signs, only from this form.

The discriminant $D = b^2 - 4ac$

The quantity under the root in the quadratic formula. Its sign alone decides the kind of roots.

Complex conjugate pair

The two roots $p + qi$ and $p - qi$ that a quadratic with real coefficients has when $D < 0$: same real part, opposite imaginary parts.

Formulas and theorems

• Definition of $i$

  $$i^2 = -1$$

  Use when Always; it is the definition. Every simplification of a power of $i$ or a root of a negative traces back to it.

  e.g. $(3i)^2 = 9i^2 = -9$.

• Powers of $i$

  $$\begin{gathered} i^1 = i, \quad i^2 = -1 \\ i^3 = -i, \quad i^4 = 1 \end{gathered}$$

  Text description

  The values 1, i, minus 1 and minus i sit a quarter turn apart around the origin, and each multiplication by i steps counterclockwise to the next, so four steps return to 1.

  Use when The cycle repeats every four: reduce $i^n$ to the power given by the remainder of $n$ on division by $4$ (remainder $0$ gives $1$).

• Square root of a negative

  $$\sqrt{-n} = i\sqrt{n}$$

  Use when $n > 0$. Convert BEFORE multiplying roots: $\sqrt{a}\cdot\sqrt{b} = \sqrt{ab}$ fails when both radicands are negative.

  e.g. $\sqrt{-12} = i\sqrt{12} = 2i\sqrt{3}$.

• Addition and subtraction

  $$\begin{gathered} (a + bi) \pm (c + di) \\ = (a \pm c) + (b \pm d)i \end{gathered}$$

  Text description

  Arrows from the origin to 3 plus i and to 1 plus 2i are two sides of a parallelogram whose far corner is their sum, 4 plus 3i.

  Use when Any two complex numbers: real parts combine with real parts, $i$ parts with $i$ parts.

  e.g. $(6 - 2i) - (3 + 4i) = 3 - 6i$: the minus reaches both parts.

• Multiplication

  $$\begin{gathered} (a + bi)(c + di) \\ = (ac - bd) + (ad + bc)i \end{gathered}$$

  Use when Any two complex numbers. In practice: expand as binomials, then replace $i^2$ with $-1$ before collecting.

  e.g. $(1 + 2i)(3 - i) = 3 - i + 6i - 2i^2 = 5 + 5i$.

• Conjugate product

  $$(a + bi)(a - bi) = a^2 + b^2$$

  Text description

  The points for 5 plus 2i and 5 minus 2i sit the same distance directly above and below the real axis, so a conjugate is the mirror image across that axis.

  Use when Any complex number; the result is real and nonnegative. A sum, not the difference of squares.

  e.g. $(5 + 2i)(5 - 2i) = 25 + 4 = 29$.

• Division

  $$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$$

  Use when $c + di \neq 0$. Multiply top and bottom by the denominator's conjugate, then finish in standard form.

• Completing the square

  $$x^2 + bx + \left(\tfrac{b}{2}\right)^2 = \left(x + \tfrac{b}{2}\right)^2$$

  Use when Leading coefficient $1$: divide the equation by $a$ first if $a \neq 1$. In an equation, add $\left(\tfrac{b}{2}\right)^2$ to both sides; in an expression, add and subtract it in the same line.

  e.g. $x^2 + 8x + 16 = (x + 4)^2$.

• The quadratic formula

  $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

  Use when $a \neq 0$, and the equation must be in standard form $ax^2 + bx + c = 0$ before reading the coefficients. The whole numerator sits over $2a$.

• The discriminant

  $$D = b^2 - 4ac$$

  Use when Coefficients read from standard form, signs included.

• Root classification by $D$

  $D > 0$: two distinct real roots. $D = 0$: one repeated real root, $x = -\tfrac{b}{2a}$. $D < 0$: a complex conjugate pair.

  Use when Standard form with real coefficients. Answers how many and what kind of roots without solving.

• Projectile height model

  $$h = -16t^2 + v_0 t + h_0$$

  Use when Height in feet, $t$ in seconds, $v_0$ the initial upward speed, $h_0$ the launch height; in metres the first term is $-4.9t^2$. Set $h = 0$ for landing, $h = H$ for a target height.

  e.g. Thrown up at $32$ ft/s from $48$ ft: $h = -16t^2 + 32t + 48$.

Problem types, step by step

Exam traps

• Trap Multiplying two negative radicands with the product rule: $\sqrt{-4}\cdot\sqrt{-9} = \sqrt{36} = 6$.

  Fix The rule fails for two negatives. Convert first: $(2i)(3i) = 6i^2 = -6$.

• Trap Leaving an $i^2$ term in a product, or treating it as $0$ or $1$.

  Fix Every $i^2$ becomes $-1$ and merges into the real part: in $(2 - i)(3 + 2i)$, $-2i^2$ becomes $+2$.

• Trap Conjugate product by the difference-of-squares reflex: $(a + bi)(a - bi) = a^2 - b^2$.

  Fix $-(bi)^2 = +b^2$, so the product is $a^2 + b^2$: a sum, always real.

• Trap Adding $\left(\tfrac{b}{2}\right)^2$ to one side of an equation, or forgetting the matching subtraction in an expression.

  Fix Equation: add it to both sides. Expression: add and subtract it in the same line.

• Trap Reading $a$, $b$, $c$ straight off $x^2 = 3x + 10$.

  Fix Coefficients are valid only from standard form: rearrange to $x^2 - 3x - 10 = 0$, then read the signs.

• Trap Sign slips inside the formula: $b = -7$ giving $b^2 = -49$, $-b$ kept as $b$, or $-4ac$ mis-signed when $c < 0$.

  Fix $b^2$ is never negative; $-b$ is the opposite of $b$; when $a$ and $c$ have opposite signs, $-4ac$ is positive, so it adds.

• Trap Calling a negative discriminant "no solution".

  Fix $D < 0$ gives a complex conjugate pair; every quadratic has roots, real or complex.

• Trap Growing each dimension by $x$ when a border or frame of width $x$ surrounds it: $(5 + x)(7 + x)$.

  Fix A border sits on both ends of each dimension, so each grows by $2x$: $(5 + 2x)(7 + 2x)$.

• Trap Auto-rejecting the negative root, or auto-keeping exactly one, in a word problem.

  Fix Test each root against the situation and say why the rejected ones go; two positive times can both be valid.