Algebra
What you must understand
Algebra at this level centers on linear and quadratic equations, systems of equations, inequalities, polynomial operations, factoring, and exponential growth. You must solve equations symbolically, interpret solutions in context, and translate word problems into equations. Know the quadratic formula, the discriminant, and the relationship between factors and roots. For systems, understand both substitution/elimination and graphical interpretation.
Traps that appear in questions
- Dividing by a variable. Dividing both sides by x discards the solution x = 0. Factor instead.
- Inequality sign flips. Multiplying or dividing by a negative reverses the inequality.
- Extraneous solutions. Squaring both sides or clearing denominators can introduce solutions that fail the original equation; always check.
- Discriminant confusion. The discriminant b² − 4ac tells the number of real roots: positive means two, zero means one, negative means none.
- Exponential vs linear growth. A constant difference means linear; a constant ratio means exponential.
Worked example
Solve x² − 5x + 6 = 0. Factor as (x − 2)(x − 3) = 0, so x = 2 or x = 3. Check the discriminant: 25 − 24 = 1 > 0, confirming two real roots. Rule of thumb: before applying the quadratic formula, try factoring and check the discriminant to predict how many solutions to expect.
Systems and context
When a system models a real situation, reject solutions that are physically impossible (negative lengths, fractional people). Interpret the intersection point as the values satisfying both conditions simultaneously.