Algebra

• ℕ — naturals: the positive whole numbers 1, 2, 3, …
• ℤ — integers: whole numbers including negatives and zero
• ℚ — rationals: anything you can write as a fraction ab (where b is not zero)
• ℝ — reals: everything on the number line
• ℝ \ ℚ — irrationals: reals that are not rational, e.g. π, √2

• Coefficient — the number multiplying the variable (the 5 in 5x)
• Variable — the letter standing for the unknown
• Power / degree — the exponent (the 3 in x³)
• Constant — a fixed number with no variable attached

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

A three-term expression ax² + bx + c factors into two brackets.

a² − b² = (a − b)(a + b)

a³ ± b³ = (a ± b)(a² ∓ ab + b²)

(a + b)² = a² + 2ab + b²

Some expressions need more than one type in turn — e.g. take out a common factor first, then use the difference of two squares on what’s left.

The discriminant b² − 4ac tells you how many real roots a quadratic has.

For ax² + bx + c, the roots add to −ba.

For ax² + bx + c, the roots multiply to ca.

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

Reverse the inequality sign whenever you multiply or divide both sides by a negative number.

|x| is the size of a number, ignoring its sign — so it is always ≥ 0.

A root that cannot be simplified to a whole number, e.g. √2.

Flip the middle sign. The conjugate of a + √b is a − √b. Their product is rational.

Remove a surd from the bottom by multiplying top and bottom by the conjugate.

Once the bases match, equate the powers and solve.

log_b(n) = p  ↔  b^p = n

log(ab) = log a + log b