Algebra: Manipulation, Equations and Inequalities
Core concepts
Algebra underpins every other topic in Mathematical Methods. You must be fluent with expanding and factorising (including difference of squares, perfect squares and grouping), simplifying algebraic fractions, index laws with rational and negative powers, and solving linear, quadratic, polynomial and simultaneous equations. For quadratics, know the discriminant $\Delta=b^2-4ac$: it tells you whether roots are real and distinct ($\Delta>0$), equal ($\Delta=0$) or non-real ($\Delta<0$). Be able to solve inequalities and express solutions in interval notation, remembering that multiplying or dividing by a negative reverses the sign. Completing the square converts $ax^2+bx+c$ into turning-point form and is also the route to the quadratic formula.
Traps
- Dividing by an expression that could be zero, thereby losing or inventing solutions.
- Forgetting that squaring both sides of an equation can introduce extraneous roots; always check.
- Mishandling negative indices: $x^{-2}=1/x^2$, not $-x^2$.
- Inequality direction errors when clearing a negative denominator.
- Reading a discriminant question as asking for the roots themselves rather than their nature.
Worked example
Solve $2x^2-5x-3=0$. Factorising: find two numbers multiplying to $2\times(-3)=-6$ and adding to $-5$, namely $-6$ and $1$. Split the middle term and factor by grouping to get $(2x+1)(x-3)=0$, so $x=-\tfrac12$ or $x=3$. Check with the discriminant: $\Delta=25+24=49>0$, confirming two distinct real roots.
Rule of thumb
Before solving, simplify and note any restrictions on the variable; after solving, substitute back to verify.