1. Algebraic expressions & indices
Rules: a^m × a^n = a^(m+n), (a^m)^n = a^(mn), a^0 =1 (a≠0), a^(−n)=1/a^n.
Example: Simplify x² × x^3 = x^5. (2x^3)/(x^2)=2x.
2. Factorization
Common techniques: take out common factor, difference of squares, perfect square trinomials, factor by grouping.
1) x^2 − 9 = (x−3)(x+3).
2) x^2 + 5x + 6 = (x+2)(x+3).
3. Linear equations
Solve 3x − 5 = 10 ⇒ 3x = 15 ⇒ x = 5.
Solve 2(3x−1)=4x+6 ⇒ 6x−2=4x+6 ⇒2x=8 ⇒ x=4.
4. Simultaneous equations (linear)
Use substitution or elimination.
Solve: x + y = 7 and 2x − y = 3. Add equations: 3x = 10 ⇒ x=10/3. Then y=7−10/3=11/3.
5. Quadratic equations
ax^2 + bx + c = 0 ⇒ x = [−b ± √(b^2 − 4ac)]/(2a)
Solve x^2 − 5x + 6 = 0 ⇒ (x−2)(x−3)=0 ⇒ x=2,3.
Solve 2x^2 + 3x − 2 = 0 ⇒ x = [−3 ± √(9 + 16)]/4 = [−3 ± 5]/4 ⇒ x=1/2 or x=−2.
6. Inequalities
When multiplying or dividing by a negative number, reverse the inequality sign.
If −2x > 6 ⇒ x < −3.
7. Algebraic fractions
Simplify by factoring numerators and denominators and cancel common factors. Solve equations by clearing denominators.
(x^2 − 1)/(x−1) = (x−1)(x+1)/(x−1) = x+1, x≠1.
8. Practice
1) Factorize x^2 + 7x + 10. Ans: (x+5)(x+2)
2) Solve system: 3x + 2y = 12, x − y = 1. Ans: Solve ⇒ x=3, y=2
3) Solve inequality: 2x − 5 ≤ 9 ⇒ x ≤ 7