Algebra

• ℕ naturals: positive whole numbers 1, 2, 3, …
• ℤ integers: whole numbers, positive or negative, including 0
• ℚ rationals: anything you can write as a fraction
• ℝ reals: everything on the number line
• ℝ \ ℚ irrationals: reals that are not rational (π, √2)

• Coefficient — the number in front (the 5 in 5x)
• Variable — the letter for the unknown
• Power / degree — the exponent (the 3 in x³)
• Constant — a number with no variable
• Terms — parts separated by + or −

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

Square the first, square the second, first by second doubled.

m, rise over run, rate of change, dy/dx, gap, and common difference — all the same number.

By terms: monomial (1), binomial (2), trinomial (3). By degree: linear (1), quadratic (2), cubic (3), quartic (4).

Take out the highest common factor first. With four terms, split into two pairs, take a common factor from each, then factor out the common bracket.

Multiply the coefficient of x² by the constant. If it's positive, find two numbers that multiply to it and add to the middle; if negative, they subtract to the middle.

a² − b² = (a − b)(a + b)

a³ ± b³ = (a ± b)(a² ∓ ab + b²)

Use more than one type in turn — take out what's common first, then factor what's left.

Top = numerator, bottom = denominator.

Factor the top and bottom fully, then cancel a common whole factor.

Get the lowest common denominator first, then add or subtract the numerators.

Cancel whole factors, never parts of a sum.

Combine the top into one fraction, combine the bottom, then flip the second and multiply.

Rearrange to get the required letter on its own.

Gather that letter's terms on one side, factor it out, then divide.

Multiply every term by the denominators to remove the fractions.

Isolate the root on one side, then square both sides.

Factor (guide number / double brackets), then let each factor = 0.

x = ( −b ± √(b² − 4ac) ) / 2a

It's in the tables — no need to learn.

b² − 4ac: > 0 two real roots; = 0 two equal roots (perfect square); < 0 complex roots.

Take half the coefficient of x, square it, and add and subtract it.

−b / a

c / a

x² − (sum)x + (product) = 0

The coefficient of x² must be 1.

Shape (a > 0 U, a < 0 ∩), roots, y-intercept and turning point.

The size of x, ignoring sign — always ≥ 0.

Solve with + and then with −.

Square both sides, then solve.

Draw the graph and read it off.

A V-shape; the vertex is where the inside = 0.

Eliminate one variable by adding or subtracting, then back-substitute.

Rearrange the linear equation, then substitute into the circle or curve.

Let a = 1/x, b = 1/y, solve, then invert.

Careful with signs, and remember to multiply the whole line by the constant.

Equate the coefficients and the constants on both sides.

If f(k) = 0, then (x − k) is a factor of f(x).

Sub in, or divide in.

Sub in values of x until you get 0 — a root is a factor of the constant.

Sub in the known root and set = 0, then solve for the unknown.

Sub in — form two equations and solve them simultaneously.

Multiply the quadratic by a made-up linear (x + k) to match the cubic.

Simple: one term, a√b. Compound: two parts, a + √b.

√(ab) = √a √b  ·  √(a/b) = √a / √b  ·  (√a)² = a

Split the number into two factors, one of which is a perfect square.

Only if the irrational part is the same; add the numbers in front.

Like algebra (FOIL); and √a · √a = a.

Change the middle sign. A surd times its conjugate is rational.

Single surd — multiply top and bottom by that surd; compound — by the conjugate.

Isolate the root, square both sides, solve, then check the answers.

Solve like equations, but flip the sign when you multiply or divide by a negative.

Split into two inequalities and solve each.

Set = 0, find the roots (smaller first), and draw a diagram for full marks.

Multiply by the denominator squared — it's ≥ 0, so the sign is safe.

Square both sides.

Use perfect squares backwards, or use the given information.

Filled dot for ≤ / ≥, open circle for < / >.

Use the rules of indices (in the tables) or the calculator to evaluate.

Get everything to the same base, then equate the powers.

Let t = a^x to form a quadratic; solve, then check the answers.

Write one unknown in terms of the other; never leave ( )= .

Turn the English into maths and define your variable.

Length × width; set up and solve the quadratic.

A length can't be negative, so throw that root out.

log_b(n) = p  ↔  b^p = n

log(ab) = log a + log b

log(a/b) = log a − log b

log aⁿ = n log a

Logs to one side, use the rules to get one log, then the definition.

Same base — drop the logs and solve.

Use the tables rule (break the bigger base down).

Bring logs in on both sides (or use BNP).