Algebra 1 formula sheet
65 formulas · what each one means · a free lesson for each
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Expressions and equations
Like terms
3x + 5x = 8x
Add the numbers in front of the same letter part.Lesson: Simplify expressions →Same to both sides
a = b → a + c = b + c, a × c = b × c
Do the same thing to both sides and they stay equal.Lesson: Linear equations →Flip the inequality
−2x < 6 → x > −3
Multiply or divide both sides by a negative: the sign turns around.Lesson: Inequalities →Proportion (cross multiply)
ab = cd → ad = bc
Top of each side times bottom of the other.Lesson: Proportions →Absolute value equation
|x| = a → x = a or x = −a
Two answers, one on each side of 0 (a must be 0 or more).Lesson: Absolute value equations →|x| > a
x < −a or x > a
Greater than: two pieces, going outwards.Lesson: Absolute value inequalities →Lines and slope
Slope
m = y2 − y1x2 − x1
Rise over run: how much y changes for each 1 of x.Lesson: Slope from two points →Slope-intercept form
y = mx + b
m is the slope, b is where the line crosses the y-axis.Lesson: Graphing y = mx + b →Point-slope form
y − y1 = m(x − x1)
The line with slope m through the point (x₁, y₁).Lesson: Point-slope form →Standard form
Ax + By = C
Find the crossing points: put x = 0, then y = 0.Lesson: Graphing Ax + By = C →Horizontal and vertical lines
y = b (slope 0) x = a (no slope)
Flat lines have slope 0; straight-up lines have no slope (undefined).Lesson: Vertical and horizontal lines →Perpendicular lines
m1 × m2 = −1 m2 = −1m1
Flip the slope and change its sign.Lesson: Parallel and perpendicular lines →Systems of two lines
cross: one answer · parallel: none · same line: every point
The answer of a system is where the lines meet.Lesson: Systems by graphing →Exponent rules
Rational exponent
amn = n√am = (n√a)m
The bottom of the power is the root, the top is the power.Lesson: Rational exponents →Scientific notation
a × 10n, 1 ≤ a < 10
One digit before the point; n counts how far the point moved.Lesson: Scientific notation →Multiplying brackets
Two brackets
(a + b)(c + d) = ac + ad + bc + bd
Each part of the first bracket times each part of the second.Lesson: Polynomials →Square of a sum
(a + b)2 = a2 + 2ab + b2
Not a² + b²: the middle 2ab is easy to forget.Lesson: Polynomials →Square of a difference
(a − b)2 = a2 − 2ab + b2
The middle part is minus, the last part is plus.Lesson: Polynomials →Factoring patterns
x² + bx + c
x2 + bx + c = (x + p)(x + q), p × q = c, p + q = b
Find two numbers that multiply to c and add to b.Lesson: Factoring →ax² + bx + c (split)
two numbers: product ac, sum b
Split bx with them, then factor by grouping.Lesson: Factoring →Grouping
ax + ay + bx + by = (a + b)(x + y)
Factor each pair, then take out the shared bracket.Lesson: Factor by grouping →Difference of squares
a2 − b2 = (a + b)(a − b)
Two squares with a minus between them.Lesson: Factoring →Perfect square
a2 + 2ab + b2 = (a + b)2
First and last are squares, middle is 2 × a × b.Lesson: Factoring completely →Difference of cubes
a3 − b3 = (a − b)(a2 + ab + b2)
Signs: same, opposite, always plus.Lesson: Sum and difference of cubes →Square roots
Product rule
√ab = √a × √b
Split a root into two roots (a, b not negative).Lesson: Multiplying square roots →Simplify
√a2 b = a√b
Take out the biggest square factor: √12 = √4 × √3 = 2√3.Lesson: Simplifying square roots →Like roots
a√c + b√c = (a + b)√c
Add only roots with the same inside.Lesson: Simplifying square roots →Two-term bottom
(a + √b)(a − √b) = a2 − b
Multiply by the partner with the other sign: the root goes.Lesson: Rationalising a two-term bottom with its partner →Quadratic equations
Standard form
ax2 + bx + c = 0
Everything on one side, 0 on the other, before you solve.Lesson: Quadratic equations →Quadratic formula
x = −b ± √b2 − 4ac2a
Works for every quadratic equation.Lesson: Quadratic equations →Discriminant
b2 − 4ac
Positive: two answers. Zero: one answer. Negative: no real answers.Lesson: The discriminant →Zero product rule
ab = 0 → a = 0 or b = 0
Why factoring solves: one of the brackets must be 0.Lesson: Solving by factoring →Completing the square
x2 + bx + (b2)2 = (x + b2)2
Add half of b, squared, to make a perfect square.Lesson: Completing the square →Sum and product of the roots
r1 + r2 = −ba r1 × r2 = ca
Read from the equation without solving.Lesson: Sum and product of the roots →Thrown ball (feet)
h = −16t2 + v0 t + h0
v₀ = starting speed, h₀ = starting height. In meters: −5t² (−4.9t²).Lesson: Quadratic word problems →Parabolas
Vertex form
y = a(x − h)2 + k
Vertex (h, k). a > 0 opens up, a < 0 opens down.Lesson: Vertex form by completing the square →Axis of symmetry
x = h
The line through the vertex that folds the parabola in half.Lesson: Parabolas →Exponential growth and decay
Exponential function
y = abx
a is the start value, b is what it is multiplied by each step.Lesson: What the numbers mean →Linear or exponential
same + each step: linear · same × each step: exponential
Look at how y changes when x goes up by 1.Lesson: Linear or exponential →Word problem formulas
Simple interest
I = Prt
Interest = principal × rate (as a decimal) × time in years.Lesson: Simple interest →Work rate
1a + 1b = 1t
a and b are each person's time alone; t is their time together.Lesson: Work-rate →Consecutive numbers
n, n + 1, n + 2 (even or odd: n, n + 2, n + 4)
Next whole number: + 1. Next even or odd: + 2.Lesson: Word problems part 1 →More formula sheets
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