AS Level Mathematics · Pure Mathematics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.
A quadratic is an expression ax2 + bx + c. Most questions ask you to rewrite it, count its roots or solve with it.
Completing the square rewrites it as a(x + p)2 + q. This form shows the turning point of the graph at once. The discriminant b2 − 4ac tells you how many real roots there are without solving. Use it whenever a question says "tangent", "repeated root", "no real roots" or "does not meet".
For an inequality, find the critical values by solving the equation, then sketch the curve to see which region is wanted. For a line and a curve, substitute the linear equation into the other one. Some equations are quadratics in disguise, in x2, √x or tan x.
When a is not 1, students subtract the wrong number. 2(x + 2)2 expands to 2x2 + 8x + 8, so you must take away 8, not 4. Always expand your answer to check it.
Answer: k < −6 or k > 6
Try each one, then open the answer.
Answer: B. Factorise: (x − 7)(x + 2) ≤ 0, critical values −2 and 7. The parabola opens upwards, so it is at or below the axis between the roots: −2 ≤ x ≤ 7.
Answer: D. Equate: x2 + 4x + 5 = 2x + c gives x2 + 2x + (5 − c) = 0. A tangent means one repeated root, so 4 − 4(5 − c) = 0, i.e. 5 − c = 1 and c = 4.
Answer: A. y = −(x2 + 4x) + 3 = −[(x + 2)2 − 4] + 3 = 7 − (x + 2)2. The square is never negative, so the maximum value 7 occurs when x + 2 = 0, i.e. at (−2, 7).