ALGEBRA FORMULAS
Algebra formulas are equalities which remain true regardless of the values of any variables which appear within it.
In our website, we have provided two calculators for algebra formulas.
One is to find the expansion for (a + b)n and other one is to find the expansion for (a - b)n.
Please click the below links to get the linear regression needed.
Expansion Calculator for (a + b)n
Expansion Calculator for (a - b)n
If you would like to have problems on algebraic identities, please click the link given below.
Worksheet on Algebraic Identities
Proving Algebraic Identity Expansion Geometrically
In this section, you will learn how to prove the expansions of algebraic identities geometrically.
Let us consider algebraic identity and its expansion given below.
(a + b)2 = a2 + 2ab + b2
We can prove the the expansion of (a + b)2 using the area of a square as shown below.
Algebraic Formulas
In this section, we are going to see the list of formulas which are being used to solve all kind of problems in the Algebra.
|
(a + b)2 = a2 + 2ab + b2 (a + b)2 = (a - b)2 + 4ab | |
|
(a - b)2 = a2 - 2ab + b2 (a - b)2 = (a + b)2 - 4ab | |
|
a2 - b2 = (a + b)(a - b) | |
|
(x + a)(x + b) = x2 + (a + b)x + ab |
Examples |
|
(a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)3 = a3 + 3ab(a + b) + b3 | |
|
(a - b)3 = a3 - 3a2b + 3ab2 - b3 (a - b)3 = a3 - 3ab(a - b) - b3 | |
|
a3 + b3 = (a + b)(a2 - ab + b2) a3 - b3 = (a - b)(a2 + ab + b2) | |
|
a3 + b3 = (a + b)3 - 3ab(a + b) a3 - b3 = (a - b)3 + 3ab(a - b) |
|
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac (a + b - c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ac (a - b + c)2 = a2 + b2 + c2 - 2ab - 2bc + 2ac (a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ac a2 + b2 = (a + b)2 - 2ab a2 + b2 = (a - b)2 + 2ab a2 + b2 = 1/2 ⋅ [(a + b)2 + (a - b)2] ab = 1/4 ⋅ [(a + b)2 - (a - b)2] (a + b + c)3 = a3 + b3 + c3 + 3a2b + 3a2c + 3ab2 + 3b2c + 3ac32 + 3bc2 + 6abc (a + b - c)3 = a3 + b3 - c3 + 3a2b - 3a2c + 3ab2 - 3b2c + 3ac2 + 3bc2 - 6abc (a - b + c)3 = a3 - b3 + c3 - 3a2b + 3a2c + 3ab2 + 3b2c + 3ac2 - 3bc2 - 6abc (a - b - c)3 = a3 - b3 - c3 - 3a2b - 3a2c + 3ab2 - 3b2c + 3ac2 - 3bc2 + 6abc (19) a² + b² = (a + b)² - 2ab (20) a² + b² = (a - b)² + 2ab (21) a² + b²=½ [(a+b)²-(a-b)²] (22) ab = ¼[(a+b)²- (a - b)²] |
(23) (a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3b²c + 3ac² + 3bc² + 6abc
(24) (a + b - c)³ = a³ + b³ - c³ + 3a²b - 3a²c + 3ab² - 3b²c + 3ac² + 3bc² - 6abc
(25) (a - b + c)³ = a³ - b³ + c³ - 3a²b + 3a²c + 3ab² + 3b²c + 3ac² - 3bc² - 6abc
(26) (a - b - c)³ = a³ - b³ - c³ - 3a²b - 3a²c + 3ab² - 3b²c + 3ac² - 3bc² + 6abc
How to remember algebra formulas with negative sign?
We can remember algebraic formula expansions like
(a + b)2, (a + b + c)2, (a + b + c)3
In the above formulas, if one or more terms is negative, how can we remember the expansion ?
This question has been answered in the following three cases.
Case 1 :
We can easily remember the expansion of (a + b + c)2.
If c is negative, then we will have
(a + b - c)2
How can we remember the expansion of (a + b - c)2 ?
It is very simple.
Let us consider the expansion of (a + b + c)2.
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
In the terms of the expansion above, consider the terms in which we find "c".
They are c2, bc, ca.
Even if we take negative sign for "c" in c2, the sign of c2 will be positive. Because it has even power 2.
The terms bc, ca will be negative. Because both "b" and "a" are multiplied by "c" that is negative.
Finally, we have
(a + b - c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ca
Case 2 :
In (a + b + c)2, if "b" is negative, then we will have
(a - b + c)2
How can we remember the expansion of (a - b + c)2 ?
It is very simple.
Let us consider the expansion of (a + b + c)2.
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
In the terms of the expansion above, consider the terms in which we find "b".
They are b2, ab, bc.
Even if we take negative sign for "b" in b2, the sign of b2 will be positive. Because it has even power 2.
The terms ab, bc will be negative. Because both "a" and "c" are multiplied by "b" that is negative.
Finally, we have
(a - b + c)2 = a2 + b2 + c2 - 2ab - 2bc + 2ca
Case 3 :
In (a + b + c)2, if both "b" and "c" are negative, then we will have
(a - b - c)2
How can we remember the expansion of (a - b - c)2 ?
It is very simple.
Let us consider the expansion of (a + b + c)2.
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
In the terms of the expansion above, consider the terms in which we find "b" and "c".
They are b2, c2, ab, bc, ac.
Even if we take negative sign for "b" in b2 and negative sign for "c" in c2, the sign of both b2 and c2 will be positive. Because they have even power 2.
The terms "ab" and "ca" will be negative.
Because, in "ab", "a" is multiplied by "b" that is negative.
Because, in "ca", "a" is multiplied by "c" that is negative.
The term "bc" will be positive.
Because, in "bc", both "b" and "c" are negative.
That is,
negative ⋅ negative = positive
Finally, we have
(a - b - c)2 = a2 + b2 + c2 - 2ab + 2bc - 2ca
In the same way, we can get idea to remember the the expansions of
(a + b - c)3, (a - b + c)3, (a - b - c)3
Apart from the stuff given above, if you would like to have problems on algebraic identities, please click the link given below.
Worksheet on Algebraic Identities
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
v4formath@gmail.com
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Recent Articles
-
Multiplying a Binomial by a Binomial
Oct 27, 20 11:40 AM
Multiplying a Binomial by a Binomial - Concept - Examples
-
Find Arithmetic Mean by Assumed Mean Method Worksheet
Oct 27, 20 10:03 AM
Find Arithmetic Mean by Assumed Mean Method Worksheet
-
Find Arithmetic Mean by Assumed Mean Method
Oct 27, 20 09:56 AM
Find Arithmetic Mean by Assumed Mean Method - Formula - Examples
