College Algebra Study Guide

Textbook Location

Property Value

Property	Value
Title	College Algebra, 2nd Edition (Teacher’s Edition)
Authors	Julie Miller & Donna Gerken
Pages	860
Size	227 MB
Location	`~/atelier/_bibliotheca/Sapientia/02_Assets/LRN-COLLEGE-ALGEBRA/textbook/Miller-Gerken-College-Algebra-2e-TE.pdf`

Title

College Algebra, 2nd Edition (Teacher’s Edition)

Authors

Julie Miller & Donna Gerken

Pages

860

Size

227 MB

Location

~/atelier/_bibliotheca/Sapientia/02_Assets/LRN-COLLEGE-ALGEBRA/textbook/Miller-Gerken-College-Algebra-2e-TE.pdf

Quick Open

zathura ~/atelier/_bibliotheca/Sapientia/02_Assets/LRN-COLLEGE-ALGEBRA/textbook/Miller-Gerken-College-Algebra-2e-TE.pdf

The Principia copy is a Git LFS pointer. Use the Sapientia copy for the actual PDF.

Study Workflow

Open textbook at the section pages (see Page Mapping)
Read the section in the PDF
Work examples by hand on paper
Fill in the corresponding section page in domus-captures
Mark status as completed when done

AsciiDoc Note-Taking Mastery

This section covers AsciiDoc patterns that make your study notes powerful, searchable, and beautiful.

Math Notation with STEM

Inline math - use \(...\) for formulas in text:

The quadratic formula is stem:[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}].

Renders as: The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Block math - use [stem] block for standalone equations:

[stem]
++++
\int_0^1 x^2 \, dx = \frac{1}{3}
++++

Renders as:

\[\int_0^1 x^2 \, dx = \frac{1}{3}\]

Collapsible Solutions

Hide answers until you’re ready to check:

.Problem 1: Solve stem:[2x + 5 = 13]
[%collapsible]
====
*Solution:*

[stem]
++++
2x + 5 = 13 \\
2x = 8 \\
x = 4
++++

*Answer:* stem:[x = 4]
====

Problem 1: Solve \(2x + 5 = 13\)

Solution:

\[2x + 5 = 13 \\ 2x = 8 \\ x = 4\]

Answer: \(x = 4\)

Learning Objective Checklists

Track what you’ve mastered:

== Learning Objectives

* [x] Define a linear equation
* [x] Solve equations with one variable
* [ ] Clear fractions from equations
* [ ] Check for extraneous solutions

Learning Objectives

Define a linear equation
Solve equations with one variable
Clear fractions from equations
Check for extraneous solutions

Definition Boxes

Make definitions stand out:

[NOTE]
.Definition: Quadratic Equation
====
A **quadratic equation** in one variable is an equation that can be written in the form:

[stem]
++++
ax^2 + bx + c = 0
++++

where stem:[a], stem:[b], and stem:[c] are real numbers and stem:[a \neq 0].
====

Definition: Quadratic Equation

A quadratic equation in one variable is an equation that can be written in the form:

\[ax^2 + bx + c = 0\]

where \(a\), \(b\), and \(c\) are real numbers and \(a \neq 0\).

Example Boxes

Structure worked examples clearly:

.Example 3 (p. 134, #27)
====
**Problem:** Find the vertex of stem:[f(x) = x^2 - 6x + 5]

**Solution:**

Complete the square:
[stem]
++++
f(x) = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4
++++

**Answer:** Vertex is stem:[(3, -4)]
====

Example 1. Example 3 (p. 134, #27)

Problem: Find the vertex of \(f(x) = x^2 - 6x + 5\)

Solution:

Complete the square:

\[f(x) = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4\]

Answer: Vertex is \((3, -4)\)

Admonition Blocks for Study Notes

TIP: When completing the square, always factor out the coefficient of stem:[x^2] first if it's not 1.

WARNING: Don't forget to check for extraneous solutions when solving rational equations!

IMPORTANT: The discriminant stem:[\Delta = b^2 - 4ac] tells you the nature of the roots.

CAUTION: Division by zero is undefined. Always check denominators!

When completing the square, always factor out the coefficient of \(x^2\) first if it’s not 1.

Don’t forget to check for extraneous solutions when solving rational equations!

The discriminant \(\Delta = b^2 - 4ac\) tells you the nature of the roots.

Division by zero is undefined. Always check denominators!

Tables for Comparing Concepts

[cols="1,2,2"]
|===
| Method | When to Use | Example

| Factoring
| When polynomial factors easily
| stem:[x^2 - 5x + 6 = 0]

| Quadratic Formula
| Always works
| stem:[2x^2 + 3x - 5 = 0]

| Completing Square
| Deriving vertex form
| stem:[x^2 + 4x + 1 = 0]
|===

Method	When to Use	Example
Factoring	When polynomial factors easily	\(x^2 - 5x + 6 = 0\)
Quadratic Formula	Always works	\(2x^2 + 3x - 5 = 0\)
Completing Square	Deriving vertex form	\(x^2 + 4x + 1 = 0\)

Method

When to Use

Example

Factoring

When polynomial factors easily

\(x^2 - 5x + 6 = 0\)

Quadratic Formula

Always works

\(2x^2 + 3x - 5 = 0\)

Completing Square

Deriving vertex form

\(x^2 + 4x + 1 = 0\)

Step-by-Step Procedures

Number your steps for algorithms:

.Procedure: Solving by Completing the Square
. Move the constant to the right side
. If stem:[a \neq 1], divide everything by stem:[a]
. Take half of stem:[b], square it: stem:[\left(\frac{b}{2}\right)^2]
. Add this to both sides
. Factor the left as a perfect square
. Take square root of both sides
. Solve for stem:[x]

Procedure: Solving by Completing the Square

Move the constant to the right side
If \(a \neq 1\), divide everything by \(a\)
Take half of \(b\), square it: \(\left(\frac{b}{2}\right)^2\)
Add this to both sides
Factor the left as a perfect square
Take square root of both sides
Solve for \(x\)

Personal Insights Section

Every section page has a "Study Notes" section. Use it for:

== Study Notes

**What clicked:**
- The connection between factoring and finding x-intercepts finally makes sense

**What needs review:**
- Still struggling with completing the square when stem:[a \neq 1]

**Connections to other topics:**
- This relates to graphing parabolas in Section 3.1

**Questions for later:**
- Why does the discriminant work? (prove it)

Keyboard Macros (experimental attribute)

Document keyboard shortcuts:

Press kbd:[Ctrl+D] to bookmark the current page in Zathura.
Press kbd:[/] to search in the PDF.

Press Ctrl+D to bookmark the current page in Zathura. Press / to search in the PDF.

Cross-References

Link between sections:

See xref:01-equations/14-quadratic-equations.adoc[Section 1.4] for the quadratic formula derivation.

Prerequisite: xref:R-prerequisites/R5-factoring.adoc[R.5 Factoring]

Quick Reference: LaTeX Math Symbols

Symbol LaTeX Renders

Symbol	LaTeX	Renders
Fraction	`\frac{a}{b}`	\(\frac{a}{b}\)
Square root	`\sqrt{x}`	\(\sqrt\{x}\)
nth root	`\sqrt[n]{x}`	\(\sqrt[n\){x}]
Exponent	`x^{n}`	\(x^{n}\)
Subscript	`a_{n}`	\(a_{n}\)
Plus/minus	`\pm`	\(\pm\)
Not equal	`\neq`	\(\neq\)
Less/greater or equal	`\leq` / `\geq`	\(\leq\) / \(\geq\)
Infinity	`\infty`	\(\infty\)
Greek letters	`\alpha, \beta, \pi`	\(\alpha, \beta, \pi\)
Absolute value	`\|x\|` or `\lvert x \rvert`	\(\lvert x \rvert\)
Set membership	`\in`	\(\in\)
Union/Intersection	`\cup` / `\cap`	\(\cup\) / \(\cap\)

Fraction

\frac{a}{b}

\(\frac{a}{b}\)

Square root

\sqrt{x}

\(\sqrt\{x}\)

nth root

\sqrt[n]{x}

\(\sqrt[n\){x}]

Exponent

x^{n}

\(x^{n}\)

Subscript

a_{n}

\(a_{n}\)

Plus/minus

\pm

\(\pm\)

Not equal

\neq

\(\neq\)

Less/greater or equal

\leq / \geq

\(\leq\) / \(\geq\)

Infinity

\infty

\(\infty\)

Greek letters

\alpha, \beta, \pi

\(\alpha, \beta, \pi\)

Absolute value

|x| or \lvert x \rvert

\(\lvert x \rvert\)

Set membership

\in

\(\in\)

Union/Intersection

\cup / \cap

\(\cup\) / \(\cap\)