College Algebra: Complex Numbers, Quadratics, and Equations Study Guide

Complex Numbers and Operations

Standard Form of Complex Numbers

Complex numbers are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined by $i^2 = -1$.

• Definition: A complex number is any number of the form $a + bi$.

• Example: $5 + 3i$ is a complex number.

• Standard Form: Always write complex numbers as $a + bi$.

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules and the property $i^2 = -1$.

• Addition/Subtraction: Combine like terms: $(a + bi) + (c + di) = (a + c) + (b + d)i$.

• Multiplication: Use distributive property and $i^2 = -1$.

• Conjugate: The conjugate of $a + bi$ is $a - bi$.

• Example: $(5 + 3i) - (-4 + i) = 9 + 2i$

• Product with Conjugate: $(3i)(5 - 2i)$ and its conjugate $(5 + 2i)$.

Division and Standard Form

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

• Formula: $rac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)}$

• Example: $rac{2i}{1 - i}$

Square Roots of Negative Numbers

Square roots of negative numbers are expressed using the imaginary unit $i$.

• Formula: $\sqrt{-a} = i\sqrt{a}$ for $a > 0$

• Example: $\sqrt{-3} - \sqrt{-121} = i\sqrt{3} - 11i = (\sqrt{3} - 11)i$

Quadratic Equations and Properties

Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula.

• Factoring: Express as product of binomials and set each factor to zero.

• Square Root Property: If $x^2 = a$, then $x = \pm \sqrt{a}$.

• Completing the Square: Transform $ax^2 + bx + c$ into $(x + d)^2 = e$.

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

• Example: $2x^2 + 2x = 16$

Perfect Square Trinomials

A perfect square trinomial is of the form $x^2 + 2bx + b^2 = (x + b)^2$.

• To complete the square: Add $\left(\frac{b}{2}\right)^2$ to $x^2 + bx$.

• Example: $x^2 + 9x$; add $\left(\frac{9}{2}\right)^2 = 20.25$.

Applications of Quadratic Equations

Projectile Motion

Quadratic equations model the height of an object in vertical motion.

• Formula: $h = h_0 + v_0 t - \frac{1}{2}gt^2$

• Example: $s = 128 + 112t - 16t^2$ models a ball thrown upward.

Area Problems

Quadratic equations can be used to solve geometric problems involving area.

• Example: If the length of a garden is 7 feet longer than its width and the area is 78 square feet, set up $w(w + 7) = 78$.

Polynomial Equations

Solving Higher-Degree Equations

Equations of degree higher than two can be solved by factoring, substitution, or other algebraic methods.

• Example: $x^3 + 10x = 7x^2$

• Substitution: For equations like $x^6 - 25x^3 - 27 = 0$, let $y = x^3$.

Radical and Rational Equations

Solving Radical Equations

Radical equations contain variables inside a root. Isolate the radical and square both sides to solve.

• Example: $\sqrt{x + 32} = x - 0$

Solving Rational Inequalities

Rational inequalities involve fractions with polynomials in the numerator and denominator. Solutions are often expressed in interval notation.

• Steps:

  1. Find critical points by setting numerator and denominator to zero.

  2. Test intervals between critical points.

  3. Express solution in interval notation.

• Example: $\frac{x - 1}{x + 6} > 0$

Summary Table: Methods for Solving Equations

Equation Type	Method	Key Formula
Quadratic	Factoring, Completing the Square, Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Complex Numbers	Add/Subtract/Multiply/Divide, Conjugate	$i^2 = -1$
Radical Equations	Isolate Radical, Square Both Sides	$\sqrt{x} = a \implies x = a^2$
Rational Inequalities	Find Critical Points, Test Intervals	Interval Notation

Additional info:

• Some problems involve application of algebraic concepts to real-world scenarios, such as projectile motion and area calculations.

• Substitution is a useful technique for solving higher-degree polynomials.

• Interval notation is used to express solutions to inequalities.