ЦТ Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Difference of squares

For all real a and b, a² − b² = (a − b)(a + b). This identity factors expressions and simplifies products such as (x − 3)(x + 3).

Rational number test

A real number is rational exactly when its decimal expansion terminates or eventually repeats. Non-terminating, non-repeating decimals, such as π and √2, are irrational.

Power rules

For positive bases: xᵐ · xⁿ = xᵐ⁺ⁿ, (xᵐ)ⁿ = xᵐⁿ, and x⁻ⁿ = 1/xⁿ. Also xᵐ/ⁿ equals the n-th root of xᵐ.

Percent change formula

A quantity changed by p percent becomes original × (1 + p/100) for an increase, or original × (1 − p/100) for a decrease. Reverse the operation to recover the original.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, solutions are x = (−b ± √(b² − 4ac))/(2a). The discriminant b² − 4ac determines whether roots are real and distinct.

Vieta's formulas

For ax² + bx + c = 0, the roots satisfy x₁ + x₂ = −b/a and x₁x₂ = c/a. These relations find root sums and products without solving.

Extraneous root check

Squaring both sides of an equation can introduce solutions that do not satisfy the original. Substitute every candidate back into the original equation and discard failures.

Absolute value equation

The equation |x| = a has solutions x = a and x = −a when a > 0, one solution x = 0 when a = 0, and no solutions when a < 0.

Absolute inequality split

For a > 0: |x| < a means −a < x < a, while |x| > a means x < −a or x > a. The inequality |x| ≤ a includes endpoints.

Sign-flip rule

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Adding or subtracting a number never reverses it.

Interval method

Factor the expression, mark all zeros on a number line, and test one point in each interval. The sign in an interval is constant between consecutive zeros.

Parabola vertex

The quadratic y = ax² + bx + c has vertex x-coordinate −b/(2a). It is a minimum when a > 0 and a maximum when a < 0.

Domain restrictions

Expressions require: denominator not zero, even-root radicand non-negative, and logarithm argument positive. Intersect all individual restrictions to obtain the full domain.

Graph transformations

Replacing f(x) by f(x) + c shifts the graph up c; f(x + c) shifts left c; −f(x) reflects across the x-axis; f(−x) reflects across the y-axis.

Pythagorean identity

For every angle x, sin²x + cos²x = 1. Dividing by cos²x gives 1 + tan²x = sec²x; dividing by sin²x gives 1 + cot²x = csc²x.

General sine solution

If sin x = a with |a| ≤ 1, then x = (−1)^k arcsin a + πk for every integer k. This captures both symmetric solution families.

Double-angle formulas

sin 2x = 2 sin x cos x. Also cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1, and tan 2x = 2tan x/(1 − tan²x).

Arithmetic progression term

The n-th term is aₙ = a₁ + (n − 1)d, where d is the common difference. The sum of the first n terms is n(a₁ + aₙ)/2.

Geometric progression sum

For bₙ = b₁qⁿ⁻¹, the first-n sum is Sₙ = b₁(qⁿ − 1)/(q − 1) when q ≠ 1. If |q| < 1, the infinite sum is b₁/(1 − q).

Geometric mean property

Three consecutive terms of a geometric progression satisfy bₙ² = bₙ₋₁ · bₙ₊₁ when all terms are positive. For arithmetic progressions, aₙ = (aₙ₋₁ + aₙ₊₁)/2.

Triangle area formulas

Area equals ½ab sin C, Heron's formula √(p(p − a)(p − b)(p − c)) with semiperimeter p, and pr where r is the inradius. Also area = abc/(4R).

Law of cosines

In a triangle, a² = b² + c² − 2bc cos A. It reduces to the Pythagorean theorem when A = 90° and finds a side or angle from the other data.

Similarity ratios

If two similar figures have linear scale factor k, their areas ratio is k² and their volumes ratio is k³. Corresponding angles remain equal.

Sphere measures

A sphere of radius r has volume 4πr³/3 and surface area 4πr². A hemisphere has half these values plus the circular base area πr².

Pyramid volume

Any pyramid or cone has volume one third of base area times perpendicular height: V = (1/3)Bh. A prism or cylinder with the same base and height has three times this volume.

Euler polyhedron formula

For a convex polyhedron, V − E + F = 2, where V counts vertices, E counts edges, and F counts faces. It checks whether a proposed polyhedron is consistent.

Combination formula

The number of k-element subsets of an n-element set is C(n, k) = n!/(k!(n − k)!). Order does not matter; permutations count ordered arrangements instead.

Complement probability

For any event A, P(not A) = 1 − P(A). This is useful when counting failures or at-least-one outcomes is easier than direct counting.

Independent events product

Events A and B are independent exactly when P(A ∩ B) = P(A)P(B). For dependent events, use P(A ∩ B) = P(A)P(B|A).

Combined work rate

If two workers complete a job in times t₁ and t₂ alone, their combined rate is 1/t₁ + 1/t₂ jobs per unit time, so combined time is t₁t₂/(t₁ + t₂).

Mixture concentration

Concentration equals amount of solute divided by total solution mass or volume. Mixing solutions conserves solute: c₁m₁ + c₂m₂ = c(m₁ + m₂).

Average speed

Average speed equals total distance divided by total time. It is not the arithmetic mean of the speeds unless equal times are spent at each speed.