Mathematics
Algebra
A quadratic equation ax²+bx+c=0 is solved through the discriminant D=b²−4ac: two real roots if D>0, one if D=0, none if D<0. Know Vieta's formulas x₁+x₂=−b/a and x₁x₂=c/a, and be able to factorise. For inequalities, multiplying or dividing by a negative number reverses the sign. The parabola y=ax²+bx+c has its vertex at x=−b/2a and opens upward when a>0. In sequences, aₙ=a₁+(n−1)d for arithmetic and bₙ=b₁qⁿ⁻¹ for geometric progressions. Probability is favourable outcomes divided by equally likely outcomes, so it always lies between 0 and 1.
Geometry
Use the Pythagorean theorem, the similarity ratio (if sides are in ratio k, areas are in ratio k²), and the circle rules: an inscribed angle equals half the central angle standing on the same arc, an angle standing on a diameter is 90°, and a tangent is perpendicular to the radius at the point of contact. Memorise sin30°=1/2, sin45°=√2/2, sin60°=√3/2, and the angle sum 180° in a triangle.
Traps
Losing the root x=0 by dividing through by x; forgetting the ± sign when taking a square root; using the side ratio instead of its square for areas; and treating a negative discriminant as an arithmetic slip instead of writing no real roots.
Worked example
Solve x²−5x+6=0. D=25−24=1, so x=(5±1)/2, giving x=2 and x=3. Check with Vieta: 2+3=5 and 2·3=6, both correct.