PRACTICE QUESTIONS IN ALGEBRAIC IDENTITIES

Problem 1 :

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

Solution :

2x + 2/x = 3

Square both sides.

(2x + 2/x)² = 3²

(2x + 2/x)(2x + 2/x) = 9

(2x)² + (2x)(2/x) + (2/x)(2x) + (2/x)² = 9

4x² + 4 + 4 + 4/x² = 9

4x² + 8 + 4/x² = 9

Subtract 8 from both sides.

4x² + 4/x² = 1

Factor.

4(x² + 1/x²) = 1

Divide both sides by 4.

x² + 1/x² = 1/4

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 :

Consider the following algebraic identity.

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

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

Substitute a + b + c = 6 and a² + b² + c² = 14.

(6)² = 14 + 2(ab + bc + ca)

36 = 14 + 2(ab + bc + ca)

Subtract 14 from both sides.

12 = 2(ab + bc + ca)

Divide both sides by 2.

6 = ab + bc + ca

The value of (a - b)² + (b - c)² + (c - a)² :

= (a - b)² + (b - c)² + (c - a)²

= a² - 2ab + b² + b²- 2bc + c² + c² - 2ca + a²

= a² + a² + b² + b² + c² + c² - 2ab - 2bc - 2ca

= 2a² + 2b² + 2c²- 2ab - 2bc - 2ca

= 2(a² + b² + c²) - 2(ab + bc + ca)

Substitute a² + b² + c² = 14 and ab + bc + ca = 6.

= 2(14) - 2(6)

= 28 - 12

= 16

Problem 3 :

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

x³ - y³ + 8(x + y)²

Solution :

Consider the square of a binomial given below.

(x - y)² = x² + y² - 2xy

Substitute x - y = 8 and xy = 5.

8² = x² + y² - 2(5)

64 + 10 = x² + y²

74 = x² + y²

Consider the square of a binomial given below.

(x + y)² = x² + y² + 2xy

Substitute x² + y² = 74 and xy = 5.

(x + y)² = 74 + 2(5)

(x + y)² = 74 + 10

(x + y)² = 84

The value of x³ - y³ + 8(x + y)² :

= x³ - y³ + 8(x + y)²

Use the identity a³ - b³ = (a - b)(a² + ab + b²).

= (x - y)(x² + xy + y²) + 8(x + y)²

Substitute.

= (8)(74 + 5) + 8(84)

=  8(79) + 672

= 632 + 672

= 1304

Problem 4 :

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

Solution :

(x + y)² = (x + y)(x + y)

(x + y)² = x² + xy + xy + y²

(x + y)² = x² + 2xy + y²

or

x² + 2xy + y² = (x + y)²

Subtract 2xy from both sides.

x² + y² = (x + y)² - 2xy

Substitute x + y = 5 and xy = 6.

x² + y² = 5² - 2(6)

x² + y² = 25 - 12

x² + y² = 13

Multiply both sides by 2.

2(x² + y²) = 2(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

Substitute a - b = 3.

3³ + 3ab(3) = 513

27 + 9ab = 513

Subtract 27 from both sides.

9ab = 486

Divide both sides by 9.

ab = 54

Problem 6 :

If a² + b² + c² = 9 and ab + bc + ca = 8, find the value of 

(a + b + c)²

Solution :

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

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

Substitute a² + b² + c² = 9 and ab + bc + ca = 8.

(a + b + c)² = 9 + 2(8)

= 9 + 16

= 25

Problem 7 :

If a + b = √7 and a - b = √5, then find the value of

8ab(a² + b²)

Solution :

a + b = √7 ----(1)

a - b = √5 ----(2)

(1) + (2) :

2a = √7 + √5

a = (√7 + √5)/2

Substitute a = (√7 + √5)/2 in (1).

(√7 + √5)/2 + b = √7

b = √7 - [(√7 + √5)/2]

b = (√7 - √5)/2

ab = [(√7 + √5)/2] ⋅ [(√7 - √5)/2]

ab = (7 - 5)/4

ab = 1/2

a² + b² = (a + b)² - 2ab

a² + b² = (√7)² - 2(1/2)

a² + b² = 7 - 1

a² + b² = 6

The value of 8ab(a² + b²) :

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)² = (a + b)² - 4ab

Substitute a - b = 4 and ab = 60.

4² = (a + b)² - 4(60)

16 = (a + b)² - 240

Add 240 to both sides.

256 = (a + b)²

or

(a + b)² = 256

Take square root on both sides.

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)² = (a - b)² + 4ab

Substitute a + b = 9m and ab = 18m².

(9m)² = (a - b)² + 4(18m²)

81m² = (a - b)² + 72m²

Subtract 72m² from both sides.

9m² = (a - b)²

or

(a - b)² = 9m²

Take square root on both sides.

a - b = √(9m²)

a - b = ±9m

Problem 10 :

Simplify :

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

Solution :

= (2345 x 2345 - 759 x 759)/(2345 - 759)

= (2345² - 759²)/(2345 - 759)

Use the identity a² - b² = (a + b)(a - b).

= (2345 + 759)(2345 - 759)/(2345 - 759)

= (3104)(1586)/(1586)

= 3104

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