Algebraic Processes
Algebraic Processes tests your fluency with symbols: expansion and factorisation, quadratic equations and their roots, simultaneous equations, linear inequalities, variation, sequences and series (AP and GP), binary operations and 2×2 matrices. NECO frequently sets a word problem to be modelled as an equation, so translating English into algebra matters as much as manipulation.
Quadratics
- Solve by factorisation, completing the square, or x = [−b ± √(b² − 4ac)]/2a.
- Discriminant b² − 4ac: positive gives two distinct real roots, zero gives a repeated root, negative gives no real roots.
- Sum of roots = −b/a; product of roots = c/a.
Simultaneous equations
Eliminate when coefficients match; substitute when one equation is linear and the other quadratic (for example y = 2x + 1 with x² + y² = 25).
Variation
Direct: y = kx. Inverse: y = k/x. Joint: y = kxz. Partial: y = kx + c. Find k first, then the required value.
Sequences and series
- AP: Tₙ = a + (n − 1)d; Sₙ = n/2[2a + (n − 1)d].
- GP: Tₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1).
Worked example
The 3rd term of an AP is 10 and the 8th term is 25. Find the sum of the first 20 terms. a + 2d = 10 and a + 7d = 25. Subtracting gives 5d = 15, so d = 3 and a = 10 − 6 = 4. Then S₂₀ = 20/2[2(4) + 19(3)] = 10(8 + 57) = 650.
Matrices
For M = [[a, b], [c, d]], det M = ad − bc and M⁻¹ = (1/det M)[[d, −b], [−c, a]], which exists only if det M ≠ 0.
Traps
- Dividing both sides by x and losing the root x = 0 in x² = 5x.
- Sign slips when b is negative in the quadratic formula: bracket the substitution.
- Using n instead of n − 1 to find Tₙ.
- Reversing an inequality sign only when multiplying or dividing by a negative number, not when adding or subtracting.