Pure Mathematics 1 begins with algebraic foundations — quadratics, polynomials, simultaneous equations, and inequalities. These skills underpin every subsequent topic in the A Level course.
Standard form: ax² + bx + c = 0. Solutions: factorisation, completing the square, quadratic formula x = (-b ± √(b²-4ac)) / 2a. Discriminant Δ = b²-4ac: Δ > 0 → two distinct real roots, Δ = 0 → one repeated root, Δ < 0 → no real roots. Completing the square: a(x + p)² + q gives vertex at (-p, q). Maximum/minimum value problems. Graph: parabola, vertex form reveals turning point.
Factor theorem: if f(a) = 0, then (x-a) is a factor. Remainder theorem: f(a) = remainder when f(x) divided by (x-a). Use to factorise cubics: find one root by trial, divide, factorise quadratic. Simultaneous equations: substitution method for one linear + one non-linear. For two linear: elimination or substitution.
Linear inequalities: solve like equations (reverse sign when multiply/divide by negative). Quadratic inequalities: solve equation first, sketch graph, read off solution set. Example: x² - 5x + 6 < 0 → (x-2)(x-3) < 0 → 2 < x < 3. Modulus (absolute value): |x - a| < b means a-b < x < a+b. |f(x)| = g(x): solve f(x) = g(x) and f(x) = -g(x), check solutions.
The discriminant b²-4ac tells you about the nature of roots without solving the equation. This is crucial for: (1) determining whether a quadratic expression is always positive/negative (if a > 0 and Δ < 0, always positive); (2) finding conditions on parameters (e.g., "find k such that the equation has equal roots" → set Δ = 0); (3) analysing intersections of curves (substitute one equation into another, examine discriminant to determine number of intersection points). It\'s one of the most commonly examined concepts at A Level.
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