How to Pass College Algebra: The Topics, a Practice Plan and the Mistakes That Cost Points

Here is how to pass college algebra: treat it as a course about functions rather than a rerun of high school algebra, practice every problem type with the book closed, mix older topics into each week's practice, check every answer by plugging it back in, and keep a running list of your own algebra slips until they stop showing up. Before each exam, take a practice test under the real rules.

So, can you pass college algebra? Most of what decides it is a set of habits you control. This guide shows what the course covers (checked against the free OpenStax textbook and three college catalogs), why it catches students who did fine in high school, a topic-by-topic plan with a free place to practice each one, the mistakes that cost the most points, and what to do before the final or if you're already behind.

Final in a few days? Go to how to pass the college algebra final exam or, for a test tomorrow, the one-day exam plan.

What college algebra covers

The course number varies (MATH 1314 at the University of Houston and UT Dallas, MATH 1111 at Abraham Baldwin Agricultural College in Georgia), but the core is consistent. The free OpenStax College Algebra 2e, written to the scope and sequence of a typical course, has nine chapters. Catalog descriptions at the University of Houston, Abraham Baldwin Agricultural College and UT Dallas list most of the same topics:

1. The algebra toolkit: exponents, radicals and rational exponents, polynomials, factoring and rational expressions. OpenStax puts these in a review chapter that some schools treat as a prerequisite and others teach.
2. Equations and inequalities: linear and quadratic equations, complex numbers, radical and absolute value equations, and inequalities.
3. Functions: notation, domain and range, rates of change, composition, transformations and inverses.
4. Polynomial and rational functions: quadratics and their vertex, polynomial graphs, dividing polynomials, finding zeros, and asymptotes.
5. Exponential and logarithmic functions: graphs, the log properties, solving exponential and log equations, and growth and decay models.
6. Systems of equations: two and three variables, often solved with matrices.

OpenStax's last two chapters, on conics and on sequences, probability and counting, show up only in some courses. ABAC's catalog calls its course "a functional approach to algebra," which is the shift to expect: fewer problems that end in x = 5, more that ask what a whole function does.

Can you pass college algebra? Why it's harder than it looks

Plenty of students who passed Algebra 2 struggle here. Four things cause most of it:

• It's about whole functions. You find a domain, describe what a graph does as x grows, build an inverse, or explain what a shift does to every point. These are questions about behavior, not single answers.
• The workload is steady. Houston's course, for example, expects regular exams and a final, online quizzes, homework on every section covered and short in-class quizzes. A missed week leaves a gap the next topic builds on.
• Small slips cost whole problems. Paul Dawkins of Lamar University, who keeps a list of common algebra errors, says he has seen every one of them in classes from algebra up to senior-level math, and blames many of them on hurrying and not paying attention to notation.
• Correct steps can give wrong answers. Radical and log equations can produce extraneous solutions, numbers that come out of valid algebra but don't satisfy the original equation. OpenStax notes these aren't mistakes in the method, so the only defense is checking (OpenStax 2.6).

The grade can matter beyond the course. At Houston, Precalculus requires College Algebra with a C- or higher, or a satisfactory placement score (MATH 2312), and Calculus 1 comes after that. If your major runs through calculus, the algebra you build now is the algebra you'll lean on later; the Calculus 1 guide shows which skills it uses most.

How to study for college algebra

Work problems with the book closed

Reading a worked example tells you whether you can follow it, not whether you can produce it. After each lecture, read one example, close the book, and solve a similar problem on a blank page. Only reopen it when you're truly stuck, and then only far enough to get moving again.

Shuffle old problem types into every week

Homework comes sorted by section, so you always know which method a problem wants. Exams don't come sorted. In two experiments, Rohrer and Taylor found that spreading practice across several sessions beat doing it all in one, and that randomly mixing problem types beat practicing them in blocks by type. On a test a week later, the mixed group's scores were "vastly superior" (Rohrer & Taylor, 2007). A later classroom study gave 126 seventh-graders the same problems over three months, either interleaved or blocked; the interleaved group scored higher on a surprise test given either 1 or 30 days later (Rohrer, Dedrick & Stershic, 2015). The authors' point applies directly: mixed practice makes you pick the method from the problem itself, which is exactly what a cumulative exam asks.

So once a week, do a short mixed set: two problems from each earlier topic, in random order, and write the method's name before you start each one.

Quiz yourself on older topics, spread out

In a precalculus course for engineering students, researchers spaced some of the retrieval practice (recalling course content from memory) well beyond each lesson and massed the rest shortly after it. Students retained the spaced material better, and those who had some spaced practice did better on the precalculus final and on exams in calculus the following semester (Hopkins et al., 2016). For you, that means a few questions from last month's topics in every study session, not a single review the night before.

Check answers by substituting them back

Every equation you solve comes with a free answer key: the original equation. Put your answer back in and see if both sides match. For inverses, check that f(f⁻¹(x)) gives x. For graphs, plug in one point. This catches extraneous solutions and most sign errors, usually in under a minute.

Keep a slip list

Write down each error you make with its cause, such as "dropped the negative when squaring" or "canceled a term, not a factor." Before every quiz, read the list. Redo each missed problem from a blank page a few days later. When the same cause appears three times, that's your next 20 minutes of practice.

How to pass college algebra: a topic-by-topic practice plan

Follow your syllabus order. Everything below is free. Most of the Lykke sections come with flashcards and a 5-question quiz (two are readings only, as marked), the demos let you change a value and watch the graph respond, and the OpenStax sections named are in the free College Algebra 2e textbook linked above, with worked examples and exercises.

A good week in any of these topics:

1. The day of the lecture, solve two or three examples with the book closed, then take the matching section's quiz.
2. Do homework without notes the first time, and add every miss to your slip list.
3. Midweek, open the demo for whatever feels like symbol-pushing. Sliding the h and k values in the transformation sandbox, for instance, shows why f(x − 3) moves right, not left.
4. At the end of the week, do a shuffled set from earlier topics and redo the slips that are due.

The college algebra mistakes that cost the most points

A lot of lost points in college algebra come from a short list of slips, and several are on Dawkins' list. Check your work for all eight:

1. Squaring a negative without parentheses. −3² is −9, because only the 3 is squared; (−3)² is 9. Type it into a calculator both ways once so you remember.
2. Subtracting only the first term. (x² + 3x − 5) − (4x − 5) is x² − x. Drop the parentheses and you get x² − x − 10.
3. Distributing halfway. 4(2x² − 10) is 8x² − 40. And in 3(2x − 5)², square first, then multiply by 3.
4. Assuming everything splits over a sum. (x + y)², √(x + y) and 1/(x + y) don't break into separate pieces. Test any rule you're unsure of with x = 1 and y = 3: (1 + 3)² is 16, but 1² + 3² is 10.
5. Canceling terms instead of factors. To simplify (3x³ − x)/x, factor the top into x(3x² − 1) first, then cancel to get 3x² − 1. You can't cancel the x out of just one term.
6. Confusing function operations. f(g(x)) is composition, not multiplication, and f(g(x)) usually differs from g(f(x)) (OpenStax 3.4). The −1 in f⁻¹(x) isn't an exponent either: 1/f(x) is the reciprocal, not the inverse (OpenStax 3.7).
7. Forgetting to flip the inequality. Multiplying or dividing both sides by a negative number reverses the symbol: −2x < 6 means x > −3.
8. Keeping an extraneous solution. Answers to radical, log and rational equations must be checked in the original equation. See the example below.

Worked example: the log equation trap

Solve log₂(x) + log₂(x − 2) = 3.

• Combine the logs with the product property: log₂(x(x − 2)) = 3.
• Rewrite in exponential form: x(x − 2) = 2³ = 8.
• Solve the quadratic: x² − 2x − 8 = 0, so (x − 4)(x + 2) = 0, and x = 4 or x = −2.
• Check both in the original equation. x = −2 would need log₂(−2), and a log of a negative number has no output, so reject it. x = 4 gives log₂(4) + log₂(2) = 2 + 1 = 3.

The answer is x = 4 only. Both numbers came from correct algebra, which is why OpenStax tells you to always check whether an answer is extraneous (OpenStax 6.6).

How to pass the college algebra final exam

Start by finding out three things: whether the final is cumulative, which calculator is allowed (graphing, scientific or none), and whether you get a formula sheet. Then practice under those rules.

About two weeks out:

1. List every problem type from the syllabus and old exams: "solve a log equation," "find the inverse," "find the asymptotes," and so on.
2. Redo your slip list first, then do shuffled sets that pull from every chapter, naming the method before you start each problem.
3. Take one full practice final on a timer with your notes closed, a week before the real one, and spend the remaining days on the problem types you missed.

The 14-day midterm plan lays out the same schedule day by day.

During the exam, skim the whole test, do the problems you're sure of first, and show every step, since partial credit needs work on the page. Before you hand it in, plug your answers back into the equations, check that each one is in the domain, and make sure word problems end with units.

If you're behind or failing college algebra

Being behind in an algebra course is fixable, but only if you work backward from the next exam rather than starting over at chapter 1:

1. Do the grade math. Find each component's weight in the syllabus, look for a dropped quiz or a final that replaces a low exam, and work out what you need. The guide to what happens if you fail a midterm walks through it.
2. Plan only for the next exam. List its topics and rebuild each with one example and three closed-book problems.
3. Backfill the toolkit only when it blocks you. If rational equations stall on factoring, spend 20 minutes on factoring, then go back.
4. Get help this week. Take your slip list to office hours, the math lab or your tutoring center. If you'd rather write first, here's how to email a professor for a meeting.
5. Decide before the withdrawal deadline. If the numbers don't work, talk to your instructor, your advisor and, if you get aid, the financial aid office. Should you drop a class? covers what a W does to your GPA and your aid.

Free college algebra resources

• OpenStax College Algebra 2e: a free textbook online or as a PDF. Use it as a second explanation when your own book's version doesn't click.
• Paul's Online Notes, Algebra: free notes from Lamar University's Paul Dawkins covering the whole course, with practice problems and the error list above.
• Khan Academy's College Algebra course: free videos and practice by unit.
• Old exams from your own course: the closest rehearsal for the real thing, so ask your instructor whether any are posted.
• Lykke's free courses: College Algebra for most topics above, and Precalculus 2e for the function families again from another angle. Explore lists the rest.