On this webpage algebra formulas, we are going to see the list of identities which are being used to solve all kind of problems in the topic Algebra in math.
Algebra formulas are equalities which remain true regardless of the values of any variables which appear within it.
Formula is the short cut of doing problems without doing the complete steps. It is very long process of doing any problem without formula. Let us consider the following example to understand the purpose of formulas in math.
Formulas |
Examples & Practice questions |
(1) (a + b)² = a² + 2 ab + b² (2) (a + b)² = (a - b)² + 4 ab | |
(3) (a - b)² = a² - 2 ab + b² (4) (a - b)² = (a + b)² - 4 ab | |
(5) a² - b² = (a + b) (a - b) | |
(6) (x+a) (x+b) =x² + (a + b) x+a b | |
(7) (a+b)³=a³+3a²b+3ab²+b³ (8) (a+b)³=a³+b³+3ab(a+b) | |
(9) (a-b)³=a³-3a²b+3ab²-b³ (10) (a-b)³=a³-b³-3ab(a-b) | |
(11) a³+b³ = (a+b)(a²-ab+b²) (12) a³+b³=(a+b)³-3 ab(a + b) | |
(13) a³-b³= (a-b)(a²+ab+ b²) (14) a³-b³=(a-b)³ +3ab(a-b) |
(15) (a+b+c)²= a²+b²+c² +2ab+2bc+2ca (16) (a+b-c)²=a²+b²+c² +2ab-2bc-2ca (17) (a-b+c)²= a²+b²+c²-2ab-2bc+2ca (18) (a-b-c)²= a²+b²+c²-2ab+2bc-2ca (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
We can remember the expansion of the identities like (a+b)² (a+b+c)², (a+b+c)³. In the above identities, if one or more terms is negative, how can we remember the expansion?
This question has been answered in the following two cases.
Case 1 :
For example, let us consider the identity of (a + b + c)²
We can easily remember the expansion of (a + b + c)².
If "c" is negative, we will have (a + b - c)²
How can we remember the expansion of (a + b - c)² ?
It is very simple.
In the terms of the expansion, a², b², c², ab, bc, ca, let us consider the terms in which we find "c"
They are c², bc, ca .
Even if we take negative sign for "c", 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" which is negative.
Finally, we have
(a + b - c)²= a² + b² + c² + 2ab - 2bc - 2ca
Case 2 :
In (a+b+c)², if both "b" and "c" are negative, we will have (a - b - c)²
How can we remember the expansion of (a - b - c)² ?
It is very simple.
In the terms of the expansion, a², b², c², ab, bc, ca, let us consider the terms in which we find "b" and "c"
They are b², c², ab, bc, ca.
Even if we take negative sign for "b" and "c", the sign of b² and c² will be positive. Because they even power "2".
The terms "ab" and "ca" will be negative.
Because, in "ab", "a" is multiplied by "b" which is negative.
Because, in "ca", "a" is multiplied by "c" which is negative.
The term "bc" will be positive.
Because, in "bc", both "b" and "c" are negative. (negative x negative = positive)
Finally, we have
(a - b - c)²= a² + b² + c² - 2ab + 2bc - 2ca
In the same we can get idea to remember the the expansions of (a + b - c)³, (a - b + c)³ (a - b - c)³
We hope that the students would have understood "How to remember algebra formulas with negative sign once they remember the expansions with positive sign".
Apart from what has been given on this web page, if you want to know more about algebraic formulas, please click here.
WORD PROBLEMS
HCF and LCM 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
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