ALGEBRA FORMULAS

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.

What are  algebra formulas?

Algebra formulas are equalities which remain true regardless of the values of any variables which appear within it.

Definition of formula

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

Examples

practice questions

(3) (a - b)² = a² - 2 ab + b²

(4) (a - b)² = (a + b)² - 4 ab

Examples

Practice Questions

(5) a² - b² = (a + b) (a - b)

Examples

Practice Questions

(6) (x+a) (x+b) =x² + (a + b) x+a b

Examples

Practice Questions

(7) (a+b)³=a³+3a²b+3ab²+b³

(8) (a+b)³=a³+b³+3ab(a+b)

Examples

Practice Questions

(9) (a-b)³=a³-3a²b+3ab²-b³

(10) (a-b)³=a³-b³-3ab(a-b)

Examples

Practice Questions

(11)  a³+b³ = (a+b)(a²-ab+b²)

(12)  a³+b³=(a+b)³-3 ab(a + b)

Examples

Practice Questions

(13) a³-b³= (a-b)(a²+ab+ b²)

(14) a³-b³=(a-b)³ +3ab(a-b)

Examples

Practice Questions

(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]

Examples

Practice Questions

(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 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  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  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. 

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ALGEBRA

Variables and constants

Writing and evaluating expressions

Solving linear equations using elimination method

Solving linear equations using substitution method

Solving linear equations using cross multiplication method

Solving one step equations

Solving quadratic equations by factoring

Solving quadratic equations by quadratic formula

Solving quadratic equations by completing square

Nature of the roots of a quadratic equations

Sum and product of the roots of a quadratic equations 

Algebraic identities

Solving absolute value equations 

Solving Absolute value inequalities

Graphing absolute value equations  

Combining like terms

Square root of polynomials 

HCF and LCM 

Remainder theorem

Synthetic division

Logarithmic problems

Simplifying radical expression

Comparing surds

Simplifying logarithmic expressions

Negative exponents rules

Scientific notations

Exponents and power

COMPETITIVE EXAMS

Quantitative aptitude

Multiplication tricks

APTITUDE TESTS ONLINE

Aptitude test online

ACT MATH ONLINE TEST

Test - I

Test - II

TRANSFORMATIONS OF FUNCTIONS

Horizontal translation

Vertical translation

Reflection through x -axis

Reflection through y -axis

Horizontal expansion and compression

Vertical  expansion and compression

Rotation transformation

Geometry transformation

Translation transformation

Dilation transformation matrix

Transformations using matrices

ORDER OF OPERATIONS

BODMAS Rule

PEMDAS Rule

WORKSHEETS

Converting customary units worksheet

Converting metric units worksheet

Decimal representation worksheets

Double facts worksheets

Missing addend worksheets

Mensuration worksheets

Geometry worksheets

Comparing  rates worksheet

Customary units worksheet

Metric units worksheet

Complementary and supplementary worksheet

Complementary and supplementary word problems worksheet

Area and perimeter worksheets

Sum of the angles in a triangle is 180 degree worksheet

Types of angles worksheet

Properties of parallelogram worksheet

Proving triangle congruence worksheet

Special line segments in triangles worksheet

Proving trigonometric identities worksheet

Properties of triangle worksheet

Estimating percent worksheets

Quadratic equations word problems worksheet

Integers and absolute value worksheets

Decimal place value worksheets

Distributive property of multiplication worksheet - I

Distributive property of multiplication worksheet - II

Writing and evaluating expressions worksheet

Nature of the roots of a quadratic equation worksheets

Determine if the relationship is proportional worksheet

TRIGONOMETRY

SOHCAHTOA

Trigonometric ratio table

Problems on trigonometric ratios

Trigonometric ratios of some specific angles

ASTC formula

All silver tea cups

All students take calculus 

All sin tan cos rule

Trigonometric ratios of some negative angles

Trigonometric ratios of 90 degree minus theta

Trigonometric ratios of 90 degree plus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 180 degree minus theta

Trigonometric ratios of 180 degree plus theta

Trigonometric ratios of 270 degree minus theta

Trigonometric ratios of 270 degree plus theta

Trigonometric ratios of angles greater than or equal to 360 degree

Trigonometric ratios of complementary angles

Trigonometric ratios of supplementary angles 

Trigonometric identities 

Problems on trigonometric identities 

Trigonometry heights and distances

Domain and range of trigonometric functions 

Domain and range of inverse  trigonometric functions

Solving word problems in trigonometry

Pythagorean theorem

MENSURATION

Mensuration formulas

Area and perimeter

Volume

GEOMETRY

Types of angles 

Types of triangles

Properties of triangle

Sum of the angle in a triangle is 180 degree

Properties of parallelogram

Construction of triangles - I 

Construction of triangles - II

Construction of triangles - III

Construction of angles - I 

Construction of angles - II

Construction angle bisector

Construction of perpendicular

Construction of perpendicular bisector

Geometry dictionary

Geometry questions 

Angle bisector theorem

Basic proportionality theorem

ANALYTICAL GEOMETRY

Analytical geometry formulas

Distance between two points

Different forms equations of straight lines

Point of intersection

Slope of the line 

Perpendicular distance

Midpoint

Area of triangle

Area of quadrilateral

Parabola

CALCULATORS

Matrix Calculators

Analytical geometry calculators

Statistics calculators

Mensuration calculators

Algebra calculators

Chemistry periodic calculator

MATH FOR KIDS

Missing addend 

Double facts 

Doubles word problems

LIFE MATHEMATICS

Direct proportion and inverse proportion

Constant of proportionality 

Unitary method direct variation

Unitary method inverse variation

Unitary method time and work

SYMMETRY

Order of rotational symmetry

Order of rotational symmetry of a circle

Order of rotational symmetry of a square

Lines of symmetry

CONVERSIONS

Converting metric units

Converting customary units

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

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

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

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 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

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

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

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6