Algebra Formula Sheet
The Algebra 1 essentials on one page: slope and the three forms of a linear equation, the quadratic formula and discriminant, distance and midpoint, the six exponent laws, and the FOIL and factoring patterns. Every formula comes with a worked example you can use as a self-check — cover the answer, run the numbers, uncover.
Slope and Equations of a Line
Slope is rise over run — how far up per step right. Use slope-intercept (y = mx + b) when a graph or the y-intercept is given; use point-slope when all you have is one point and the slope — it skips the algebra of hunting down b.
| Rule | Formula | Worked example |
|---|---|---|
| Slope Through two points | m = (y₂ − y₁)/(x₂ − x₁) | (0, 1), (2, 5) → m = (5 − 1)/(2 − 0) = 2 |
| Slope-intercept form Equation of a line | y = mx + b | m = 2, b = 1 → y = 2x + 1 (at x = 4, y = 9) |
| Point-slope form Through a point with slope m | y − y₁ = m(x − x₁) | m = 2, (3, 7) → y − 7 = 2(x − 3), which is y = 2x + 1 |
Quadratics
The quadratic formula solves any quadratic — factoring is faster when it works, the formula always works. Check the discriminant first: it tells you how many real roots exist before you compute anything.
| Rule | Formula | Worked example |
|---|---|---|
| Quadratic formula Roots of ax² + bx + c = 0 | x = (−b ± √(b² − 4ac))/(2a) | x² − 5x + 6 = 0 → x = (5 ± 1)/2 → x = 2 or x = 3 |
| Discriminant Number of real roots | D = b² − 4ac | x² − 4x + 4 → D = 16 − 16 = 0 → one real root (x = 2) |
Coordinate Geometry
The distance formula is the Pythagorean theorem wearing coordinates — the two legs of the right triangle are the horizontal and vertical gaps. The midpoint is just averaging: x with x, y with y.
| Rule | Formula | Worked example |
|---|---|---|
| Distance Between two points | d = √((x₂ − x₁)² + (y₂ − y₁)²) | (1, 2), (4, 6) → √(3² + 4²) = √25 = 5 |
| Midpoint Between two points | M = ((x₁ + x₂)/2, (y₁ + y₂)/2) | (1, 2), (4, 6) → M = (2.5, 4) |
Exponent Laws
Every law below needs the same base to apply. The memory hook: multiplying adds exponents, dividing subtracts them, a power raised to a power multiplies them — and a negative exponent never makes the value negative, it flips the factor to the bottom.
| Rule | Formula | Worked example |
|---|---|---|
| Product rule Same base, multiply | xᵃ · xᵇ = xᵃ⁺ᵇ | 2³ · 2⁴ = 2⁷ = 128 |
| Quotient rule Same base, divide | xᵃ / xᵇ = xᵃ⁻ᵇ | 2⁵ / 2² = 2³ = 8 |
| Power of a power Raise a power to a power | (xᵃ)ᵇ = xᵃᵇ | (2³)² = 2⁶ = 64 |
| Power of a product Exponent distributes | (xy)ᵃ = xᵃyᵃ | (2 · 5)² = 2² · 5² = 100 |
| Zero exponent Any nonzero base | x⁰ = 1 (x ≠ 0) | 7⁰ = 1 |
| Negative exponent Flip, do not negate | x⁻ⁿ = 1/xⁿ | 2⁻³ = 1/2³ = 1/8 |
Multiplying Binomials and Factoring
FOIL is how you expand; the three patterns below are how you factor — running FOIL backwards on sight. The one mistake worth drilling out: (a + b)² is a perfect square, not a² + b² — check with numbers, (1 + 2)² = 9 but 1² + 2² = 5.
| Rule | Formula | Worked example |
|---|---|---|
| FOIL Multiply two binomials | (a + b)(c + d) = ac + ad + bc + bd | (x + 3)(x + 2) = x² + 5x + 6 |
| Square of a sum Perfect-square trinomial | (a + b)² = a² + 2ab + b² | (x + 3)² = x² + 6x + 9 |
| Square of a difference Perfect-square trinomial | (a − b)² = a² − 2ab + b² | (x − 4)² = x² − 8x + 16 |
| Difference of squares Factor on sight | a² − b² = (a + b)(a − b) | x² − 9 = (x + 3)(x − 3) |
A reference sheet that doubles as a quiz
Algebra formulas fail students in a specific way: not by being forgotten, but by being half-remembered — the quadratic formula with a missing 2a, point-slope with the x and y swapped. Reading a formula list does not catch those errors; computing with one does. That is what the example column is for, and why this page pairs each rule with numbers a student can verify with a pencil in thirty seconds.
The sections follow the Algebra 1 year in order. Lines come first because everything later is built on them — the distance formula is the Pythagorean theorem in coordinate clothing, and the quadratic formula is where factoring stops being enough. The exponent laws and factoring patterns at the bottom are the ones that carry into Algebra 2, which is why they earn space even on a one-page sheet. Whether it is called a formula sheet, cheat sheet or reference sheet, the job is the same: one place to glance instead of ten pages of notes.
Teaching tips
Check the discriminant before computing
Before running the quadratic formula, compute b² − 4ac: positive means two real roots, zero means one, negative means none exist. On a multiple-choice question that is the whole answer in one line of arithmetic.
Verify FOIL with the middle term
Expanding (x + 3)(x + 2), most errors land in the middle. The check: the middle coefficient should be the sum of the two numbers (3 + 2 = 5) when the leading term is x². Wrong middle term means the outer and inner products were added wrong.
Drill the (a + b)² trap with numbers
The most common algebra error in the wild: (a + b)² treated as a² + b². Numbers expose it instantly — (1 + 2)² is 9, but 1² + 2² is 5. Any student who has seen that pair once stops making the mistake.
Negative exponents flip, they do not negate
2⁻³ is 1/8, not −8. The exponent moves the factor across the fraction bar and that is all it does. This one shows up on every standardized test, usually wearing a variable: x⁻² = 1/x².
Frequently asked questions
What is on an algebra formula sheet?
Slope from two points, slope-intercept and point-slope forms, the quadratic formula, the discriminant, the distance and midpoint formulas, the six exponent laws (product, quotient, power of a power, power of a product, zero and negative exponents), and the binomial patterns — FOIL, perfect squares and the difference of squares.
Who is this algebra reference sheet for?
Algebra 1 students (grade 8-9) are the core audience. It also works as an end-of-course review sheet, a summer-refresher reference before Algebra 2, and a parent-facing cheat sheet for homework help.
When should I use the quadratic formula instead of factoring?
Try factoring first when the leading coefficient is 1 and small — it is faster when it works. Switch to the formula when factoring stalls, when roots are fractions or irrational, or when the question only asks how many real roots exist (then the discriminant alone answers it).
What is the difference between slope-intercept and point-slope form?
Both describe the same line. Slope-intercept (y = mx + b) is the readable one — it shows the slope and y-intercept directly, which is why graphing uses it. Point-slope (y − y₁ = m(x − x₁)) is the working one — write it immediately from any point and the slope, then simplify into slope-intercept if needed.
Is this the same as a cheat sheet?
Same content, honest name: cheat sheet usually means a personal reference for studying, and most teachers allow reference sheets on algebra quizzes precisely because the skills being taught are using the formulas, not recalling them. This page is for studying and homework — during tests, use whatever the teacher provides.