🔎 WHY IT WORKSWhy completing the square finds the min: (x − h)² is a square, so it is never negative — its smallest value is 0, when x = h. So a(x − h)² + k is smallest (a > 0) when the bracket is 0, giving y = k. That is why the turning point is (h, k).

🔎 WHY IT WORKSWhy b² − 4ac counts the roots: in the formula x = [−b ± √(b² − 4ac)]/2a, the ± gives two answers only if the square root is a real non-zero number. If b² − 4ac > 0 → 2 roots; = 0 → the ± adds nothing → 1 (equal) root; < 0 → you cannot square-root a negative → no real roots. For a line and a curve, substituting gives a quadratic whose roots are the x-values where they meet.

🔎 WHY IT WORKSWhy the conjugate removes the surd: (a + √b)(a − √b) = a² − (√b)² = a² − b — this is the difference of two squares, and squaring √b leaves a whole number. So multiplying top AND bottom by the conjugate clears the surd from the denominator without changing the value (you are multiplying by 1).

        2x² + 3x − 2
x − 3 ) 2x³ − 3x² − 11x + 6
        2x³ − 6x²
              3x² − 11x
              3x² −  9x
                   −2x + 6
                   −2x + 6
                         0

🔎 WHY IT WORKSWhy the remainder theorem works: division always gives f(x) = (x − a) × Q(x) + R. Put x = a: the bracket becomes 0, so the whole first part vanishes and f(a) = R. If R = 0, then (x − a) divides exactly — it is a factor.

🔎 WHY IT WORKSWhy cover-up works: partial fractions reverse adding fractions. Multiply both sides by the denominator and you get an identity true for EVERY x — so choose x-values that make brackets 0: each one "kills" all terms except one, leaving one unknown at a time.

🔎 WHY IT WORKSWhy ⁿCᵣ appears: (a + b)ⁿ = (a + b)(a + b)…(a + b). To make the term with bʳ you pick b from r of the n brackets and a from the rest. The number of ways to choose those r brackets is ⁿCᵣ — that is the coefficient.

🔎 WHY IT WORKSWhy the log laws work: a log just asks "what power?" — log₂8 = 3 because 2³ = 8. Multiplying numbers ADDS their powers (2³ × 2⁴ = 2⁷), so log(xy) = log x + log y. Dividing subtracts powers, and a power multiplies them — every log law is an index law in disguise.

🔎 WHY IT WORKSSymmetry: the point for 180° − θ is the mirror image of θ in the y-axis — same height (same sin), opposite x (cos flips sign). That is why sin(180° − θ) = sin θ. Same idea gives 180° + θ (both flip) and 360° − θ (only y flips).

🔎 WHY IT WORKSWhy period = 360°/b: sin bx repeats when bx goes through 360°, so x only needs to go through 360° ÷ b. For sin(x/b), x must go through 360° × b. Max = c + |a|, min = c − |a| because sin only ranges −1 to 1.

🔎 WHY IT WORKSWhere sin² + cos² = 1 comes from: on the unit circle the point (cos θ, sin θ) is 1 unit from O, so by Pythagoras cos²θ + sin²θ = 1. Divide every term by cos²θ → 1 + tan²θ = sec²θ. Divide by sin²θ → cot²θ + 1 = cosec²θ. That is all three given identities from one picture.

🔎 WHY IT WORKSDouble angle comes free: put B = A in sin(A + B) = sin A cos B + cos A sin B → sin 2A = 2 sin A cos A. Put B = A in cos(A + B) → cos 2A = cos²A − sin²A, then use sin² + cos² = 1 to get the other two forms. You don't need to memorise them separately — they're on the sheet anyway.

🔎 WHY IT WORKSWhy m₁m₂ = −1: turning a line 90° swaps the "rise" and "run" and flips one sign: gradient a/b becomes −b/a. Multiply them: (a/b)(−b/a) = −1. A circle equation is just Pythagoras: every point (x, y) is distance r from the centre (a, b), so (x − a)² + (y − b)² = r².

🔎 WHY IT WORKSWhy taking lg makes a straight line: y = axⁿ is a curve. Take lg of both sides: lg y = lg a + n lg x. If you call Y = lg y and X = lg x, this is Y = nX + lg a — exactly y = mx + c. So the gradient is n and the intercept is lg a.