# PRACTICE QUESTIONS IN ALGEBRAIC IDENTITIES

Problem 1 :

If 2x + (2/x)  =  3, what is the value of x2 + 1/x2 ?

Solution :

Given :

2x + (2/x)  =  3

Taking squares on both sides.

[2x + (2/x)]2  =  32

(2x)2 + 2 (2x) (2/x) + (2/x)2  =  9

4x2 + 4 + 4/x2  =  9

4x2 + 4/x2  =  5

4(x2+1/x2)  =  5

(x2+1/x2)  =  5/4

Problem 2 :

If a+b+c  =  6 and a2+b2+c2  =  14, what is the value of

(a-b)2 + (b-c)2 + (c-a)2 ?

Solution :

(a-b)2 + (b-c)2 + (c-a)2

By finding the expansion, we get

=  a2-2ab+b2 + b2-2bc+c2+c2-2ab+a2

=  2a2 + 2b+ 2c2-2(ab+bc+ca)

=  2(a2+b2+c2)-2(ab+bc+ca)  ---------(1)

(a+b+c)2  =  a2+b2+c2+2ab+2bc+2ca

By applying the given values, we get

62  =  14+2(ab+bc+ca)

22  =  2(ab+bc+ca)

ab+bc+ca  =  11

By applying this value in (1), we get

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

=  2(6)+2(11)

=  12+22

=  34

Problem 3 :

If x-y  =  8 and xy  =  5, what is the value of

x3-y3+8(x+y)2

Solution :

(x+y)=  x2+y2+2xy   -----(1)

(x-y)2  =  x2+y2-2xy   -----(2)

By applying the given values in (2), we get

82  =  x2+y2-2(5)

64+10  =  x2+y2

x2+y =  74

By applying the value of x2+y2 in (1), we get

(x+y)2  =  74+2(5)  ==>  84

x3-y=  (x-y)3 +  3xy(x-y)

x3-y3  =  83 +  3(5)(8)  ==>  632

x3-y3+8(x+y) =  632 + 8(84)

=  512+672

=  1304

Problem 4 :

If x + y  =  5 and xy  =  6 and x > y, then find 2(x2+y2)

Solution :

Given :

x + y  =  5 and xy  =  6

x2+y2  =  (x+y)2-2xy

x2+y2  =  52-2(6)

=  25 - 12

x2+y2  =  13

2(x2+y2)  =  26

Problem 5 :

If a3-b3  =  513 and a-b  =  3, what is the value of ab ?

Solution :

a3-b3  =  513

(a-b)3+3ab(a-b)  =  513

33+3ab(3)  =  513

27+9ab  =  513

9ab  =  486

ab  =  54

Problem 6 :

If a2+b2+c2  =  9 and ab+bc+ca  =  8,

what is the vale of (a+b+c)2

Solution :

Given :

a2+b2+c2  =  9, ab+bc+ca  =  8

(a+b+c)2  =  a2+b2+c2+2ab+2bc+2ca

(a+b+c)2  =  a2+b2+c2+2(ab+bc+ca)

(a+b+c)2  =  9+2(8)

=  9+16

(a+b+c)2  =  25

Problem 7 :

a+b  =  √7 and a-b  =  √5,

then find the value of 8ab(a2+b2)

Solution :

Given :

a+b  =  √7 ----(1)

a-b  =  √5  ----(2)

(1)+(2)

2a  =  √7+√5

a  =  (√7+√5)/2

By applying the value of a in (1), we get

b  =  √7 - [(√7+√5)/2]

b  =  (√7-√5)/2

ab  =  ((√7+√5)/2) ⋅ (√7-√5)/2

ab  =  (7-5)/4

ab  =  1/2

a2+b=  (a+b)2-2ab

a2+b2  =  (√7)2-2(1/2)

a2+b2  =  6

8ab(a2+b2)  =  8(1/2) (6)

8ab(a2+b2)  =  24

Problem 8 :

If a-b  =  4 and ab  =  60, what is the value of a+b ?

Solution :

a-b  =  √(a+b)2-4ab

4  =  √(a+b)2-4(60)

16  =  (a+b)2-240

16+240  =  (a+b)2

(a+b) =  256

a+b  =  √256

a+b  =  ±16

Problem 9 :

If a+b  =  9m and ab  =  18m2, what is the value of a-b ?

Solution :

a-b  =  √(a+b)2-4ab

a-b  =  √(9m)2-4(18m2)

a-b  =  √81m2-72m2

a-b  =  √9m2

a-b  =  ±9m

Problem 10 :

Simplify :

(2345 x 2345 - 759x759)/(2345-759)

Solution :

(2345 x 2345 - 759x759)/(2345-759)

By comparing the given question with the formula

a2-b2  =  (a+b)(a-b)

Let a  =  2345  and b  =  759

=  (23452 - 7592)/(2345-759)

=  (2345+759)(2345-759)/(2345-759)

=  2345+759

=  3104 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

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

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 