← Algebraic methods

Simultaneous, exponents & logs

Substitute linear into quadratic; match bases for exponents. Exam-style y = x + 1 and x² + y² = 25 Find all intersection points. Substitute: x² + (x + 1)² = 25 → 2x² + 2x − 24 = 0 x² + x − 12 = 0 → (x + 4)(x − 3) = 0 → x = −4, 3 Points: (−4, −3) and (3, 4) ✓ Both pairs — a line can cut a circle twice. ✕

A "let n = …" substitution needs the negative root checked

Equations like 5(5²ˣ) − 5(5ˣ) − 120 = 0 become quadratic when you let n = 5ˣ, but a base like 5ˣ or 4ˣ can never be negative — so after solving the quadratic for n, ALWAYS check whether a negative root is possible before taking logs of it. Papers ask this every year and the excluded root is exactly where a mark hides.

Practice for this standard covers the whole topic — flashcards and quiz.