Basic Mathematics

Co-ordinate Geometry
Distance Formula

Section Formula

Mid Point Formula

Centroid Formula

Incentre Formula

Area Formulas

Ratios
If a/b = c/d THEN ;

Alternendo

d/b = c/a

Invertendo

b/a = d/c

Componendo

(a+b)/b = (c+d)/d

Dividendo 

(a-b)/b = (c-d)/d

Componendo - Dividendo

(a+b)/(a-b) = (c+d)/(c-d)

Theorem on Equal Ratios

(a+c)/(b+d) = (a-c)/(b-d) = k

Expansions
(a+b)2= a2 + b2 + 2ab

(a-b)2= a2+ b2- 2ab

a2 - b2= (a-b)(a+b)

(a+b)3 = a3+ 3a2b +3ab2+b3

(a-b)3 = a3- 3a2b + 3ab2 - b3

a3+ b3= (a+b)(a2-ab+b2)

a3 - b3= (a - b)(a2+ab+b2)

(a+b+c)2 = a2 + b2+c2 + 2(ab+bc+ac)

a2 + b2 = (a + b)2- 2ab

a2 + b2 = (a - b)2 + 2ab

(a - b)2 = (a + b)2- 4ab

Identity
f(x) = g(x) is an identity if both have the same value for all real values of x.

Polynomials
1) First order polynomial : f(x) = 0 ;

e.g. ax + b = 0

x = -b /a (only one solution)

2) Second Order Polynomial : ax2 + bx + c = 0

x = [-b ± √(b2-4ac)] / 2a  (two solutions)

3) Third Degree Polynomial :

Can be solved if the equation can be factorized into linear and quadratic sub-equations.

4) nthDegree Polynomial : has n solutions

Exponential Equations
e.g.   af(x) = ag(x), where a =/ 1

then f(x) = g(x)

Equate the exponents to find the solution.

If it cannot be put in the form, substitute another variable y so as to get a polynomial equation.

Logarithmic Equations
if  logaf(x) = logag(x) ; then f(x) = g(x). f(x) and g(x) should be positive.

if the above form cannot be achieved, put a variable y , so as to get a polynomial equation.

Functions
Every function can be plotted on graph. Based on the behavior of graphs, we have an important section of Mathematics , called the Calculus , which studies the following :- If a function is defined as max{ 2x + 5, 4x + 7} ; it means that our function is the outer boundary amongst the to given functions.
 * 1) x and corresponding y co-ordinates
 * 2) Continuity of graph of a function
 * 3) Rate of change of y co-ordinate with respect to x co-ordinates
 * 4) Revival of function from rate of change of y w.r.t. x.

Some Important Functions
A Function is an expression which defines the behaviour of numbers.

Modulus Function : f(x) = |x-a|

= x-a   x>a

= a-x   x", "<" instead of "=" is called an inequation.

Laws of inequations

1) a + b > a + c ; then b > c

2) if a > b         ; then ca > cb

3) if ab > ac ; then b > c  ;  if a > 0

4) if a > 0, b > 0 ; then a+b > 0 , ab > 0

if a < 0, b < 0 ; then a+b < 0 , ab > 0

if a<0, b >0 ; then ab > 0

5) ax > ay  ; then x > y if a > 1

6) logax > logay  ; then x > y if a > 1

x < y if 0 < a < 1

for evaluating roots of an inequations, use basic rules discussed in the chapter earlier. The answer always lies as an inequation.