|ALGEBRA|O Level Mathematics Comprehensive Formula and Algebra Study Guide

Mensuration Formulas

Area of a right-angled triangle: $\frac{1}{2} \times \text{base} \times \text{height}$
Area of a circle: $\pi \times r^2$
Circumference of a circle: $2 \times \pi \times r$
Area of a parallelogram: $\text{base} \times \text{height}$
Area of a trapezium: $\frac{1}{2} \times (a + b) \times h$
Volume of a cube: $l^3$
Volume of a pyramid: $\frac{1}{3} \times \text{base area} \times \text{height}$
Volume of a cone: $\frac{1}{3} \times \pi \times r^2 \times h$
Volume of a prism: $\text{base area} \times \text{length}$
Volume of a sphere: $\frac{4}{3} \times \pi \times r^3$
Volume of a hemisphere: $\frac{2}{3} \times \pi \times r^3$
Surface area of a sphere: $4 \times \pi \times r^2$
Surface area of a hemisphere (curved): $2 \times \pi \times r^2$
Total surface area of a hemisphere: $3 \times \pi \times r^2$
Curved surface area of a cone: $\pi \times r \times l$
Total surface area of a cone: $\pi \times r \times l + \pi \times r^2$
Volume of a cylinder: $\pi \times r^2 \times h$
Curved surface area of a cylinder: $2 \times \pi \times r \times h$
Total surface area of a cylinder: $2 \times \pi \times r \times h + 2 \times \pi \times r^2$

Trigonometry and Geometry

Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
Cosine Rule: $a^2 = b^2 + c^2 - 2 \times b \times c \times \cos(A)$
Area of an arbitrary triangle: $\frac{1}{2} \times a \times b \times \sin(C)$
Pythagoras Theorem: $a^2 + b^2 = c^2$
Gradient of a line ( $m$ ): $m = \frac{y_2 - y_1}{x_2 - x_1}$
Length of a line: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Equation of a line: $y = mx + c$ or $y - y_1 = m(x - x_1)$

Algebraic Basics and Expansion

Expansion involves multiplying to remove brackets.
Simplification involves adding or subtracting like terms.
Example 1 (Basic Expansion): $5(2a + 5) = 10a + 25$
Example 2 (Distribution with Negatives): $5 - 2(a - 3)$ $= 5 - 2a + 6$ $= 11 - 2a$
Example 3 (Multiple Terms): $3(2x + y) - (x + 3y)$ $= 6x + 3y - x - 3y$ $= 5x$
Example 4 (Binomial Expansion): $(2a - 3)(3a + 2)$ $= 6a^2 + 4a - 9a - 6$ $= 6a^2 - 5a - 6$
Example 5 (Perfect Squares): $(2x - 4)^2$ $= (2x - 4)(2x - 4)$ $= 4x^2 - 8x - 8x + 16$ $= 4x^2 - 16x + 16$
Example 6 (Triple Brackets): $(a - 3)(a + 2)(a - 4)$ $= (a - 3)[a^2 - 4a + 2a - 8]$ $= (a - 3)(a^2 - 2a - 8)$ $= a^3 - 2a^2 - 8a - 3a^2 + 6a + 24$ $= a^3 - 5a^2 - 2a + 24$

Factorization Techniques

Factorizing is the process of writing an expression as a product of its factors using brackets.

1. Taking Out Common Factors

Example: $10a + 25 = 5(2a + 5)$
Example: $3a + 6ab + 9a^2b = 3a(1 + 2b + 3ab)$
Example: $3(a - 2) + 2b(a - 2) = (a - 2)(3 + 2b)$

2. Box / Cross Method (for Quadratic Expressions)

Example: $x^2 + 3x + 2 = (x + 2)(x + 1)$
Example: $3a^2 + 15a - 18 = 3(a^2 + 5a - 6) = 3(a + 6)(a - 1)$

3. Grouping (Taking out common factor twice)

Example: $ax - bx + 2ay - 2by$ $= x(a - b) + 2y(a - b)$ $= (a - b)(x + 2y)$
Example: $12mr - 4ms - 5ns + 15nr$ $= 4m(3r - s) - 5n(s - 3r)$ $= 4m(3r - s) + 5n(3r - s)$ $= (3r - s)(4m + 5n)$

4. Difference of Two Squares

General Formula: $a^2 - b^2 = (a - b)(a + b)$
Example: $a^2 - 25 = a^2 - 5^2 = (a - 5)(a + 5)$
Example: $81x^2 - 100y^2 = (9x)^2 - (10y)^2 = (9x - 10y)(9x + 10y)$
Example: $m^2 - 1 = (m + 1)(m - 1)$

List of Perfect Squares

${1^2=1}$ , ${2^2=4}$ , ${3^2=9}$ , ${4^2=16}$ , ${5^2=25}$ , ${6^2=36}$ , ${7^2=49}$ , ${8^2=64}$ , ${9^2=81}$ , ${10^2=100}$ , ${11^2=121}$ , ${12^2=144}$

Algebraic Fractions

1. Simplification

Example: $\frac{12a^3b}{48a^2b^2} = \frac{a}{4b}$
Example: $\frac{4x^2 - 16}{3x^2 + x - 10} = \frac{4(x - 2)(x + 2)}{(3x - 5)(x + 2)} = \frac{4(x - 2)}{3x - 5}$

2. Addition and Subtraction

Addition Example: $\frac{2}{x + 1} + \frac{3}{x - 2} = \frac{2(x - 2) + 3(x + 1)}{(x + 1)(x - 2)} = \frac{2x - 4 + 3x + 3}{(x + 1)(x - 2)} = \frac{5x - 1}{(x + 1)(x - 2)}$
Subtraction Example: $\frac{2}{x + 2} - \frac{1}{x + 3} = \frac{2(x + 3) - (x + 2)}{(x + 2)(x + 3)} = \frac{2x + 6 - x - 2}{(x + 2)(x + 3)} = \frac{x + 4}{(x + 2)(x + 3)}$

3. Multiplication and Division

Multiplication Example: $\frac{5x^3}{a} \times \frac{3x^2}{8} = \frac{15x^5}{8a}$
Division Example: $\frac{5x^3}{a} \div \frac{8}{3x^2} = \frac{5x^3}{a} \times \frac{3x^2}{8} = \frac{15x^5}{8a}$

Equations and Formulas

Quadratic Formula

For $ax^2 + bx + c = 0$ : $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Solving Equations

Linear Example: $5x + 1 = 11 \rightarrow 5x = 10 \rightarrow x = 2$
Linear Example with Fractions: $\frac{x}{5} - \frac{2x - 3}{4a} = -3$ Multiply by LCM (20) to clear denominators: $4x - 5(2x - 3) = -3 \times 20$ $4x - 10x + 15 = -60$ $-6x = -75$ $x = 12.5$
Quadratic Example (Factored): $(x - 2)(x + 3) = 0 \rightarrow x = 2 \text{ or } x = -3$
Quadratic Example (Using Formula): $2x^2 - 15x + 13 = 0$ $x = \frac{15 \pm \sqrt{(-15)^2 - 4(2)(13)}}{2(2)} = \frac{15 \pm \sqrt{121}}{4}$ $x = 6.5 \text{ or } x = 1$
Perfect Square Completion: $(x - 5)^2 - 8 = 0 \rightarrow (x - 5)^2 = 8 \rightarrow x - 5 = \pm \sqrt{8}$ $x = 7.83 (3 \text{ s.f.}) \text{ or } x = 2.17 (3 \text{ s.f.})$

Changing the Subject of a Formula

Example 1: $t = \sqrt{4s + 1}$ . To make $s$ the subject: $t^2 = 4s + 1$ $t^2 - 1 = 4s$ $s = \frac{t^2 - 1}{4}$
Example 2: $v = am + 2an$ . To make $a$ the subject: $v = a(m + 2n)$ $a = \frac{v}{m + 2n}$
Example 3: $P = \frac{RT}{v - b}$ . To make $v$ the subject: $P(v - b) = RT$ $Pv - Pb = RT$ $Pv = RT + Pb$ $v = \frac{RT + Pb}{P}$
Example 4: $v^2 = u^2 + 2as$ . To make $u$ the subject: $v^2 - 2as = u^2$ $u = \pm \sqrt{v^2 - 2as}$

Mathematics in Practical Situations

Simple Interest: $I = \frac{P \times R \times T}{100}$
Compound Interest: $A = P(1 + \frac{r}{100})^n$
Percentage change: $\frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%$
$n^{th}$ term formula for a number sequence with common difference $d$ : $T_n = a + (n - 1)d$
Direct Proportion formula: $y = k \times x$
Inverse Proportion formula: $y = \frac{k}{x}$