Algebra
Core concepts
Algebra rewards structural thinking: manipulating expressions, factoring, solving linear and quadratic equations and inequalities, working with exponents and radicals, and handling systems of equations. You should be able to complete the square, apply the quadratic formula, use the discriminant to count real roots, and recognise when substitution or elimination is faster than direct solving. Absolute value equations and inequalities, rational expressions, and equations with extraneous roots (squaring both sides) are standard.
Traps to watch
- Dividing by an expression that may be zero loses solutions; factor instead.
- Squaring both sides can introduce extraneous roots - always check in the original equation.
- Multiplying an inequality by a negative number reverses the direction.
- For x^2 = k with k > 0, there are two solutions, not one.
- Domain restrictions (denominators, even roots) must be stated and respected.
Rule of thumb
When a problem mentions a sum and a product of two unknowns, reach for Vieta's formulas: for x^2 - Sx + P = 0, the roots sum to S and multiply to P. This often bypasses solving entirely.
Worked example
Solve 2/(x-1) + 3/(x+2) = 1. Multiply through by (x-1)(x+2), noting x ≠ 1, -2: 2(x+2) + 3(x-1) = (x-1)(x+2). Left side: 5x + 1. Right side: x^2 + x - 2. So x^2 - 4x - 3 = 0, giving x = 2 ± sqrt(7). Both satisfy the domain restriction and the original equation.