TMUA Quadratics

A quadratic on TMUA Paper 1 is almost never asked at face value. The test pulls in one extra condition — equal roots, parallel tangent, integer solutions, no intersection, minimum value zero — and asks you to recover the parameter that makes the condition hold. The skill being measured is whether you can translate the condition into a clean algebraic statement, then solve.

Three families of move show up in roughly equal proportion across recent papers:

• Discriminant conditions. Equal roots, distinct real roots, or no real roots translate to $b^2 - 4ac$ equalling, exceeding, or falling below zero.
• Vertex form. Completing the square gives the minimum value, the line of symmetry, and the range in one rearrangement.
• Line-meets-curve. Substituting a line into a quadratic produces another quadratic; tangent when its discriminant vanishes, secant when positive.

The move. A condition on the quadratic’s output (min value, range, touch-the-axis) is usually a discriminant condition. A condition on the roots (sum, product, integer) is usually Vieta. Sort that out before reaching for algebra.