Master algebra — the language of mathematics — with this essential flashcard deck. Algebra is the gateway to all higher mathematics, and every topic from geometry to calculus builds directly on algebraic thinking. This deck covers everything from the basics of...
Master algebra — the language of mathematics — with this essential flashcard deck. Algebra is the gateway to all higher mathematics, and every topic from geometry to calculus builds directly on algebraic thinking. This deck covers everything from the basics of linear
equations to quadratic functions, factoring, exponent laws, logarithms, and systems of equations.
Whether you are preparing for your GCSE, SAT, ACT, A-Level, or first-year university math course, this deck gives you the rules, formulas, and techniques you need to solve problems confidently and quickly. All content is based on publicly available educational
resources including OpenStax Algebra and Khan Academy.
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Flip through all 28 cards in interactive study mode.
x = (−b ± √(b²−4ac)) / 2a — used to solve ax² + bx + c = 0. The discriminant (b²−4ac): > 0 = two real roots, = 0 = one real root, < 0 = no real roots.
Δ = b² − 4ac in a quadratic equation. Δ > 0 → two distinct real roots. Δ = 0 → one repeated real root. Δ < 0 → no real roots (two complex roots).
y = mx + b where m = slope (rise/run) and b = y-intercept (where the line crosses the y-axis).
y − y₁ = m(x − x₁) where m is the slope and (x₁, y₁) is a known point on the line. Useful when you know the slope and one point.
Ax + By = C where A, B, and C are integers and A ≥ 0. Useful for finding intercepts: set x=0 for y-intercept, y=0 for x-intercept.
m = (y₂ − y₁) / (x₂ − x₁) — rise over run. Positive slope = upward, negative = downward, zero = horizontal, undefined = vertical line.
Parallel lines: same slope (m₁ = m₂). Perpendicular lines: slopes are negative reciprocals: m₁ × m₂ = −1 (e.g., slopes of 2 and −½).
Product: aᵐ × aⁿ = aᵐ⁺ⁿ. Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Power of power: (aᵐ)ⁿ = aᵐⁿ. Zero exponent: a⁰ = 1. Negative exponent: a⁻ⁿ = 1/aⁿ.
a⁻ⁿ = 1/aⁿ. It means the reciprocal. Example: 2⁻³ = 1/2³ = 1/8. A negative exponent does NOT make the number negative — it flips it.
aᵐ/ⁿ = ⁿ√(aᵐ) — the denominator is the root, the numerator is the power. Example: 8²/³ = (³√8)² = 2² = 4.
a² − b² = (a + b)(a − b). Example: x² − 9 = (x+3)(x−3). Only works when both terms are perfect squares and separated by subtraction.
a² + 2ab + b² = (a+b)² and a² − 2ab + b² = (a−b)². Example: x² + 6x + 9 = (x+3)². Recognize them when the constant is (b/2)².
For x² + bx + c = 0: 1. Move c to right side. 2. Add (b/2)² to both sides. 3. Factor left side as (x + b/2)². 4. Take square root of both sides. Example: x²+6x=7 → (x+3)²=16 → x=1 or x=−7.
A method to multiply two binomials: First, Outer, Inner, Last. Example: (x+2)(x+3) = x²+3x+2x+6 = x²+5x+6.
A relation where each input (x) has exactly one output (y). Written as f(x). The vertical line test determines if a graph is a function — a vertical line hits the curve at most once.
(f ∘ g)(x) = f(g(x)) — apply g first, then apply f to the result. Example: if f(x)=x² and g(x)=x+1, then f(g(x)) = (x+1)². Order matters: f∘g ≠ g∘f in general.
A function f⁻¹(x) that reverses f(x). If f(a)=b, then f⁻¹(b)=a. To find: swap x and y, then solve for y. The graph is a reflection of f(x) across the line y=x.
logₐ(b) = c means aᶜ = b. The logarithm answers: "what power must a be raised to, to get b?" Example: log₂(8) = 3 because 2³ = 8.
ln(x) = logₑ(x) — logarithm with base e ≈ 2.718 (Euler's number). ln(eˣ) = x and e^(ln x) = x. ln(1) = 0 and ln(e) = 1.
1. Solve one equation for one variable. 2. Substitute that expression into the other equation. 3. Solve for the remaining variable. 4. Substitute back to find the first variable.
1. Multiply equations to make one variable's coefficients equal. 2. Add or subtract equations to eliminate that variable. 3. Solve for the remaining variable. 4. Substitute back.
aₙ = a₁ + (n−1)d where aₙ = nth term, a₁ = first term, d = common difference. Sum of n terms: Sₙ = n/2 × (a₁ + aₙ). Example: 2, 5, 8, 11… d=3.
aₙ = a₁ × rⁿ⁻¹ where r = common ratio. Sum of n terms: Sₙ = a₁(1−rⁿ)/(1−r) (r≠1). Infinite sum (
(a+b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ for k=0 to n. Coefficients come from Pascal's Triangle. Example: (a+b)² = a² + 2ab + b². (a+b)³ = a³ + 3a²b + 3ab² + b³.
Same rules as equations EXCEPT: when multiplying or dividing by a negative number, flip the inequality sign. Example: −2x < 6 → x > −3.
d = √((x₂−x₁)² + (y₂−y₁)²) — derived from the Pythagorean theorem. Gives the straight-line distance between (x₁,y₁) and (x₂,y₂).
M = ((x₁+x₂)/2, (y₁+y₂)/2) — the average of the x-coordinates and the average of the y-coordinates.
