Number and Algebra
Number and Algebra
The Number and Algebra domain tests whether you can move fluently between expressions, equations, and graphs, and justify each algebraic step. Expect simplifying expressions with indices, expanding and factorising, rearranging formulae, solving linear and quadratic equations, solving simultaneous equations, and interpreting inequalities. The exam rewards structure: show the line of reasoning, not just the answer.
Core rules and concepts
- Index laws: (a^m a^n=a^{m+n}), (a^m/a^n=a^{m-n}), ((a^m)^n=a^{mn}), (a^0=1) for (a\ne0).
- Expand carefully; factorising reverses expansion. For a quadratic (ax^2+bx+c), find two numbers with product (ac) and sum (b); split the middle term and group.
- Solving equations: do the same operation to both sides. For inequalities, multiplying or dividing by a negative number reverses the inequality sign.
- Simultaneous equations: elimination is often safest; substitution is useful when one variable is already isolated.
- Check solutions by substitution; a quadratic can have two, one, or no real solutions.
Worked example
Solve (3x^2-5x-2=0).
Find factors of (3\times(-2)=-6) that add to (-5): (-6) and (1). Split the middle term: (3x^2-6x+x-2=0). Group: (3x(x-2)+1(x-2)=0), so ((3x+1)(x-2)=0). Therefore (3x+1=0) or (x-2=0), giving (x=-\frac13) or (x=2). Check (x=2): (3(4)-10-2=0). Check (x=-\frac13): (3(\frac19)+\frac53-2=\frac13+\frac53-2=0).
Traps that cost marks
- Dropping a negative sign when expanding or factorising.
- Dividing both sides by a variable and losing a solution; factorise instead.
- Forgetting to flip an inequality when multiplying or dividing by a negative.
- Rounding too early; keep exact values until the final line.
- Giving only one solution to a quadratic without justification.