Algebra 1 Review: Key Topics and Formulas to Prepare for Your Final
An Algebra 1 final covers linear equations and slope, systems, exponents, polynomials and factoring, quadratics, functions, and inequalities. Review each topic in order, keep the core formulas on one sheet, and redo a quick problem whenever you hesitate. Use the sections below to locate a weak topic and then open its full guide.
What This Review Covers
This page is a topic-and-formula review, not a full course. It lists the seven areas most Algebra 1 finals test, gives the formula you need for each, and shows one short refresher problem. If you want the whole course laid out as units and lessons, the Algebra 1 study guide is the course catalog. Vocabulary definitions and a practice set are separate pages, so this review stays focused on what to memorize and redo.
Linear Equations and Slope
A linear equation graphs as a straight line. Its slope measures rise over run:
m = (y2 - y1) / (x2 - x1)Write the line in slope-intercept form
y = mx + bwhere m is the slope and b is the y-intercept. Given one point (x1, y1) and the slope, use point-slope form:
y - y1 = m(x - x1)Refresher: find the line through (2, 3) and (4, 7). The slope is (7 - 3)/(4 - 2) = 2, so y = 2x + b; using (2, 3) gives 3 = 4 + b, so b = -1 and the line is y = 2x - 1.
A rising line has positive slope, a falling line negative, a horizontal line 0, and a vertical line undefined.
Systems of Equations
A system of two linear equations has one solution where the lines cross, no solution when the lines are parallel, or infinitely many solutions when the equations are identical.
Use substitution when one equation is already solved for a variable. For
y = 2x - 13x + y = 9substitute 2x - 1 for y:
3x + (2x - 1) = 95x - 1 = 9x = 2then y = 2(2) - 1 = 3. The solution is (2, 3), the intersection point.
Use elimination when matching coefficients are easy to make opposite. The systems guide walks through both methods step by step.
Exponent Rules
Three rules cover most Algebra 1 exponent work:
a^m · a^n = a^(m + n) producta^m / a^n = a^(m - n) quotient(a^m)^n = a^(m·n) powerAlso keep a^0 = 1 and a^(-n) = 1/a^n. Refresher: simplify (x^3 · x^4) / x^2. The numerator is x^(3+4) = x^7, then divide by x^2 to get x^5.
When the base has several factors, apply each rule factor by factor. The exponents guide covers negative and zero exponents in depth.
Polynomials and Factoring
Factoring rewrites a polynomial as a product. Always pull out the greatest common factor first, then look for a pattern.
The difference of squares is the most-used pattern:
a^2 - b^2 = (a + b)(a - b)Refresher: factor x^2 - 9. Since 9 = 3^2, this is x^2 - 3^2 = (x + 3)(x - 3).
For a trinomial x^2 + bx + c, find two numbers that multiply to c and add to b. Factoring is the key prerequisite for solving quadratics and simplifying rational expressions.
Quadratic Equations
A quadratic has the form ax^2 + bx + c = 0 with a ≠ 0. When factoring is not quick, use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)The expression under the square root, b^2 - 4ac, is the discriminant. It tells you the number of real roots: positive means two, zero means one repeated root, negative means none.
Refresher: solve x^2 - 5x + 6 = 0. Factoring gives (x - 2)(x - 3) = 0, so x = 2 or x = 3. The quadratic equations guide compares factoring, completing the square, and the formula.
Functions, Tables, and Graphs
A function assigns exactly one output to each input. On a graph, the vertical line test confirms that no vertical line crosses the curve more than once. The domain is the set of allowed inputs; the range is the set of outputs.
Read a table as paired (input, output) values. If two rows share the same input with different outputs, the relation is not a function.
The functions guide develops notation, domain, and graphs in full.
Inequalities
Solving an inequality is like solving an equation, with one rule: when you multiply or divide both sides by a negative number, flip the direction of the sign.
For
-2x + 6 < 10subtract 6: -2x < 4. Divide by -2 and flip:
x > -2Graph the result on a number line with an open circle at -2 and shading to the right.
The Algebra 1 Formula Sheet
Topic | Formula to remember |
Slope | m = (y2 - y1) / (x2 - x1) |
Line | y = mx + b |
Point-slope | y - y1 = m(x - x1) |
Product of powers | a^m · a^n = a^(m + n) |
Quotient of powers | a^m / a^n = a^(m - n) |
Difference of squares | a^2 - b^2 = (a + b)(a - b) |
Quadratic formula | x = (-b ± √(b^2 - 4ac)) / (2a) |
Two-Week Final Review Checklist
Work through this list and mark every topic where a refresher problem still trips you up.
- Linear: Find slope and write the line through two points.
- Systems: Solve one system by substitution and one by elimination.
- Exponents: Simplify a product-quotient-power chain.
- Factoring: Factor out the GCF and apply the difference of squares.
- Quadratics: Solve by factoring, then check the discriminant.
- Functions: Run the vertical line test and read a table.
- Inequalities: Solve one inequality and watch the negative-flip rule.
Common Mistakes
- Forgetting that a vertical line has undefined slope, not zero slope.
- Flipping a sign when multiplying an inequality by a positive number.
- Canceling terms across addition instead of factoring first.
- Forgetting a ≠ 0 in the quadratic formula.
- Treating parallel lines as having one solution.
- Stopping after factoring out the GCF and missing a second pattern.
- Confusing the domain (inputs) with the range (outputs).
Where This Review Fits
This review consolidates the topics and formulas; each section points to the guide that explains the method in depth. Start from the Algebra 1 study guide if you need the course order, then come back here before the final.
Practice Algebra 1 with Gauth
Pick the topic your checklist marked weak. Open Gauth Math Courses, select Create Course, and enter a focused request such as "factoring and the quadratic formula" plus your level. After one short activity, redo a fresh review problem on paper before comparing answers.
Sources and Review
- OpenStax Elementary Algebra 2e, accessed October 14, 2026.