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subject: Formula Chart For Algebra [print this page]

Introduction to formula chart for algebra

A combination of constants and variables form algebraic expression.

The highest power of the variable is called the degree of the variable.

An algebraic expression with powers is called polynomial.

A polynomial can be linear, quadratic, or cubic.

We have certain formulae for quadratic and cubic expressions

Formula Chart for Quadratic Expressions:-

The following quadratic formulae helps in solving the algebra problems very fast.

(a+b)^2= a^2 + 2ab + b^2 for example 102 2 can be written as (100+2)^2= 1002+2(100)(2)+ 22=10,000+400+4=10404

(a-b)^2 = a^2 -2ab + b^2 here 982 is written as (100- 2)^2 = 1002- 2(100)(2) + (2)^2= 10,000-400 + 4 = 9,604

a^2 - b^2 = ( a+b)(a-b) for example 102 - 72 = (10+7)(10-7) = 17 times 3 = 51

Next let us take 3 variables and find the square of these trinomials.

Let us denote them as 'a' , 'b' and 'c'

(a+b+c)^2 = a^2 +b^2+ c^2 +2ab +2bc+2ac

suppose one of them is negative, then the formula makes a modification

For example (a+b-c)^2 = a^2+b^2+c^2+2ab -2bc-2ca

Now let us work out an intriguing problem.

simplify 178+x 178 + 2x178 x 122 +122x122

If you start multiplying these numbers , it will take a long time

So let us use the formula

let a=178 and b= 122

then a^2 = 178 x178 and b^2 = 122 x 122 .

We also notice 2x178x122 represents 2ab

so those numbers fit into the formula a^2+2ab + b^2 which is nothing but (a+b)^2 that is (178+122)^2

What is 178+122 add them. You get 300.

So square 300

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We get 300 x 300 = 90,000.

Wasn't it easy to use formula here?

Formula Chart for Cubic Expressions:-

(a +b)^3= a^3 + b^3 +3a^2b +3ab^2

(a-b)^3 = a^3 - b^3 - 3a^2b + 3ab^2

Let us evaluate (1005)^3 Using formula we can break it into (1000+5)^3 = 10003+ 53+3(1000)^2(5) + 3(1000)(5)^2

= 1,000,000,000 +125 +15,000,000+75,000

= 1,015,075,125

next (997)^3 can be evaluated using formula (1000-3)^3 = 001,026,973

Next there is another set of cubic formulae.

a^3 - b^3 = (a-b)( a^2 + ab +b^2) example 103 - 93 = (10-9)(102 +(10)(9) + 92= (1)(100 +90 +81) = (1)(2710 = 271

a^3 + b^3 = ( a + b) (a^2 - ab + b^2)

Next we have a^3+ b^3+ c^3 - 3abc = (a+b+c) (a^2+ b^2+2- ab - bc - ca)

If (a+b+c)= 0 then the formula reduces to a^3 + b^3 +c^3 = 3abc

For example what is 103 - 73 - 33 Here a=10 b= -7 c= -3

so a + b + c = 10-7-3 = 0

therefore 103+(-7)^3+ (-3)^3 = 3 (10)(-7)(-3) = 3 x10 x21 = 630

The big problem just became easy to solve

by: mathqa

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