Algebra
Algebra tests simplification and factorization of expressions, linear and quadratic equations, indices, surds, and inequalities. SEE problems range from direct simplification to word problems that require forming an equation before solving it.
Core rules
- Indices: a^m×a^n=a^(m+n); a^m÷a^n=a^(m−n); (a^m)^n=a^(mn); a^0=1 (a≠0); a^(−n)=1/a^n; a^(1/n)=ⁿ√a.
- Factorize by common factor, difference of squares a²−b²=(a+b)(a−b), and splitting the middle term.
- Quadratic equation ax²+bx+c=0: factorize or use x=[−b±√(b²−4ac)]/(2a). Check roots in the original equation.
- Surds: rationalize 1/(√a−√b) by multiplying by (√a+√b)/(√a+√b).
- Inequality: adding/subtracting keeps the sign; multiplying or dividing by a negative reverses it.
Worked example
Solve x²−5x+6=0. Split the middle term: x²−2x−3x+6=0, so x(x−2)−3(x−2)=0, giving (x−2)(x−3)=0. Hence x=2 or x=3. Substituting both into the original equation gives 0=0, so neither is extraneous. For 1/(√5−2), multiply by (√5+2)/(√5+2): denominator 5−4=1, so the simplified form is √5+2.
Traps
- Reversing an inequality only when multiplying/dividing by a negative - not when adding.
- Cancelling terms across a sum, e.g. writing (x+3)/x as 3.
- Losing the negative root of a quadratic.
- Adding surds as √a+√b=√(a+b), which is false.
- Forgetting to state restrictions such as x≠0 or a≠0.