Algebra

Only combine like terms — same letter, same power. So 5a + 3a = 8a, but a and b stay apart. Remember x² = x·x.

A number or letter outside a bracket multiplies everything inside. For two brackets, multiply each term of the first by the whole second bracket, then collect.

Like signs give +, unlike signs give −. A plus is ‘I have’, a minus is ‘I owe’.

(a + b)² means (a + b)(a + b) — write it out twice, then expand. Never just square each part.

Take out the highest common factor: 4x + 6y = 2(2x + 3y),  6a² + 8ab = 2a(3a + 4b).

Group in pairs, factor each, then the common bracket comes out: x² + 3x + ax + 3a = (x + 3)(x + a).

a² − b² = (a − b)(a + b), e.g. x² − 49 = (x − 7)(x + 7).

If a product = 0, one factor = 0. Factorise, then set each bracket to 0.

To factorise ax²+bx+c: find two numbers that multiply to a×c and add to b. Split the middle term, then group.

Get everything to one side = 0, factorise, set each bracket to 0. The answers are the roots.

When it won’t factorise: x = (−b ± √(b²−4ac)) / 2a.

Roots p and q → (x−p)(x−q) = 0.

Put everything over one common bottom, multiply out each top, then tidy. Watch a minus in front of a bracket: −(x − 7) = −x + 7.

The common denominator is the two brackets multiplied. Leave the bottom as a product like (x + 1)(2x + 3).

Factorise top and bottom, then cancel a matching bracket: (x + 7)² / ((x − 7)(x + 7)) = (x + 7)/(x − 7).

Multiply out any brackets, gather the letter on one side and the numbers on the other, then divide.

Cross-multiply (fraction = fraction) or multiply every term by the common denominator, then solve.

Call the unknown a letter, turn each sentence into an equation, then solve.

Undo the operations: multiply out, gather the wanted letter, factor it out, divide. Cross-multiply to clear a fraction.

Square both sides to remove a root. But √(a+b) ≠ √a + √b.

Match the number in front of one letter, then add or subtract to eliminate it. Clear any fractions first.

Two lines cross at the solution. For word problems, name two letters and write two equations.

Solve like an equation, but flip the sign when you multiply or divide by a negative. List only the values in the given set.

ℕ naturals 1, 2, 3, …   ℤ integers …, −1, 0, 1, …   ℚ rational (fractions)   ℝ real.

Filled dot = included (≤, ≥); open dot = not included (<, >).

Divide the leading terms, multiply back, subtract, bring down — then repeat until nothing is left. Put in 0x² for any missing term.

Equal gaps → linear. Equal gaps-of-the-gaps → quadratic. Multiply by a constant → exponential. Use T_n to reach any term.