Algebra Essentials
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.
All Cards in This Deck
Question
What is the quadratic formula?
Answer
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.
Question
What is the discriminant and what does it tell you?
Answer
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).
Question
What is the slope-intercept form of a line?
Answer
y = mx + b where m = slope (rise/run) and b = y-intercept (where the line crosses the y-axis).
Question
What is the point-slope form of a line?
Answer
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.
Question
What is the standard form of a linear equation?
Answer
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.
Question
How do you find the slope between two points?
Answer
m = (y₂ − y₁) / (x₂ − x₁) — rise over run. Positive slope = upward, negative = downward, zero = horizontal, undefined = vertical line.
Question
What are parallel and perpendicular slopes?
Answer
Parallel lines: same slope (m₁ = m₂). Perpendicular lines: slopes are negative reciprocals: m₁ × m₂ = −1 (e.g., slopes of 2 and −½).
Question
What are the laws of exponents?
Answer
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.
Question
What is a negative exponent?
Answer
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.
Question
What is a fractional exponent?
Answer
aᵐ/ⁿ = ⁿ√(aᵐ) — the denominator is the root, the numerator is the power. Example: 8²/³ = (³√8)² = 2² = 4.
Question
What is the difference of squares factoring formula?
Answer
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.
Question
What are perfect square trinomials?
Answer
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.
Question
How do you complete the square?
Answer
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.
Question
What is the FOIL method?
Answer
A method to multiply two binomials: First, Outer, Inner, Last. Example: (x+2)(x+3) = x²+3x+2x+6 = x²+5x+6.
Question
What is a function in mathematics?
Answer
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.
Question
What is function composition?
Answer
(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.
Question
What is an inverse function?
Answer
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.
Question
What is a logarithm?
Answer
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.
Question
What are the log rules?
Answer
Product: log(xy) = log(x)+log(y). Quotient: log(x/y) = log(x)−log(y). Power: log(xⁿ) = n·log(x). Change of base: logₐ(x) = log(x)/log(a).
Question
What is the natural logarithm (ln)?
Answer
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.
Question
How do you solve systems of equations by substitution?
Answer
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.
Question
How do you solve systems of equations by elimination?
Answer
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.
Question
What is the arithmetic sequence formula?
Answer
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.
Question
What is the geometric sequence formula?
Answer
aₙ = a₁ × rⁿ⁻¹ where r = common ratio. Sum of n terms: Sₙ = a₁(1−rⁿ)/(1−r) (r≠1). Infinite sum (
Question
What is the binomial theorem?
Answer
(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.
Question
What are inequalities and how do you solve them?
Answer
Same rules as equations EXCEPT: when multiplying or dividing by a negative number, flip the inequality sign. Example: −2x < 6 → x > −3.
Question
What is the distance formula between two points?
Answer
d = √((x₂−x₁)² + (y₂−y₁)²) — derived from the Pythagorean theorem. Gives the straight-line distance between (x₁,y₁) and (x₂,y₂).
Question
What is the midpoint formula?
Answer
M = ((x₁+x₂)/2, (y₁+y₂)/2) — the average of the x-coordinates and the average of the y-coordinates.