Algebra study guide
This guide is designed to help you understand the core ideas that commonly appear in introductory college Algebra courses. Read the explanation, follow the example, then try the practice questions without looking back. The goal is to understand the reasoning—not just memorize an answer.
Course requirements vary by school and instructor. Use this as a learning and review resource alongside your assigned textbook, lectures, and instructor guidance.
Linear equations & inequalities
Concept: A linear equation says two expressions are equal and the variable has power 1. Solve by doing the same inverse operation to both sides until the variable is isolated. For inequalities, the process is similar, but multiplying or dividing by a negative number reverses the inequality sign.
Worked example: Solve 3x + 5 = 20: subtract 5 to get 3x = 15, then divide by 3. So x = 5. For −2x > 8, divide by −2 and reverse the sign: x < −4.
Remember: Keep both sides balanced; combine like terms first; reverse an inequality only when multiplying or dividing by a negative.
Common mistakes: Changing only one side of the equation; forgetting to distribute; forgetting to reverse an inequality sign.
Practice it
- Solve 4x − 7 = 21.
- Solve 5 − 2x ≤ 11.
- Write an inequality for “a number is at least 12.”
Functions and graphing
Concept: A function assigns exactly one output to each input. Function notation f(x) means “the output when the input is x.” Graphs show how inputs and outputs relate; slope measures rate of change and intercepts show where a graph crosses an axis.
Worked example: If f(x)=2x−3, then f(4)=2(4)−3=5. The graph is a line with slope 2 and y-intercept −3.
Remember: Use the vertical-line test; identify domain and range; read slope as rise/run; substitute carefully into function notation.
Common mistakes: Treating f(x) as multiplication; mixing x- and y-intercepts; reading graph scales incorrectly.
Practice it
- For f(x)=3x+1, find f(−2).
- Does x²+y²=9 define y as a function of x?
- Find the slope through (1,2) and (5,10).
Systems of equations
Concept: A system asks for values that satisfy two or more equations at the same time. Graphing finds the intersection, substitution replaces one variable with an equivalent expression, and elimination adds or subtracts equations to cancel a variable.
Worked example: x+y=10 and x−y=2. Add the equations: 2x=12, so x=6. Substitute back: y=4. The solution is (6,4).
Remember: A solution must satisfy every equation; choose substitution when a variable is already isolated; choose elimination when coefficients line up.
Common mistakes: Stopping after finding one variable; sign errors during elimination; forgetting to check the ordered pair.
Practice it
- Solve x+y=9 and x−y=3.
- Solve 2x+y=7 and x−y=2.
- What does “no solution” look like on a graph?
Polynomials & factoring
Concept: A polynomial is a sum of terms with whole-number exponents. Factoring rewrites a polynomial as a product. Start by removing the greatest common factor, then look for patterns such as trinomials or difference of squares.
Worked example: Factor x²+5x+6. Find two numbers that multiply to 6 and add to 5: 2 and 3. So x²+5x+6=(x+2)(x+3).
Remember: Factor out the GCF first; for x²+bx+c find factors of c that add to b; a²−b²=(a−b)(a+b).
Common mistakes: Forgetting a common factor; choosing factors with wrong signs; confusing adding exponents with multiplying powers.
Practice it
- Factor x²+7x+12.
- Factor 3x²−12x.
- Factor x²−25.
Quadratic equations
Concept: A quadratic has highest power 2. You can solve by factoring, completing the square, graphing, or the quadratic formula. The solutions are the x-values where the parabola crosses the x-axis.
Worked example: Solve x²−5x+6=0. Factor: (x−2)(x−3)=0. By the zero-product property, x=2 or x=3.
Remember: Set the equation equal to zero before factoring; the discriminant b²−4ac tells how many real roots exist; check both roots.
Common mistakes: Factoring before moving all terms to one side; losing the ± in the quadratic formula; sign errors with b.
Practice it
- Solve x²−9=0.
- Solve x²+6x+5=0.
- For x²+2x+5=0, what does the discriminant say?
Exponents & radicals
Concept: Exponent rules simplify repeated multiplication. When multiplying like bases, add exponents; when dividing, subtract; a negative exponent means reciprocal. Radicals undo powers, and square roots can often be simplified by pulling out perfect-square factors.
Worked example: x³·x⁴=x⁷. Also √72=√(36·2)=6√2.
Remember: Apply exponent rules only to like bases; x⁰=1 for x≠0; √(ab)=√a√b for nonnegative a,b.
Common mistakes: Adding exponents when terms are added instead of multiplied; treating √(a+b) as √a+√b; forgetting negative exponents create reciprocals.
Practice it
- Simplify a⁵/a².
- Simplify (x³)².
- Simplify √50.