WORKSHEET ON ALGEBRAIC IDENTITIES

Problem 1 :

Expand :

(5x + 3)² 

Problem 2 :

If a - b = 3 and a² + b² = 29, find the value of ab.

Problem 3 :

Find the value : 

[√2 + 1/√ 2]²

Problem 4 :

If (x + a)(x + b)(x + c) = x³ - 10x² + 45 x - 15, then find the value

a² + b² + c²

Problem 5 :

If 2x + 3y = 13 and xy = 6, then find the value of 

8x³ + 27y³

Problem 6 :

Factor : 

27x³ + 64y³

Problem 7 :

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

Problem 8 :

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

(a - b)² + (b - c)² + (c - a)² ?

Problem 9 :

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

x³ - y³ + 8(x + y)²

Problem 10 :

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

Answers

1. Answer :

(5x + 3)² is in the form of (a + b)².

(a + b)²  =  a² + 2ab + b²

Substitute a = 5x and b = 3.

(5x + 3)²  =  (5x)² + 2 ⋅ 5x ⋅ 3 + 3²

=  25x² + 30x + 9

2. Answer :

Given a - b = 3 and a² + b² = 29.

To find the value of ab, we can use the following algebraic identity. 

(a - b)²  =  a² - 2ab + b²

or

(a - b)²  =  a² + b² - 2ab

Substitute a - b  =  3 and a² + b²  =  29.

3²  =  29 - 2ab

9  =  29 - 2ab

Add 2ab to each side. 

9 + 2ab  =  29

Subtract 9 from each side.

2ab  =  20

Divide each side by 2. 

ab  =  20

3. Answer :

[√2 + 1/√ 2]² is in the form of (a + b)².

(a + b)²  =  a² + 2ab + b²

Substitute a  =  √2 and b  =  1/√2.

[√2 + 1/√ 2]²  =  (√2)² + 2 ⋅ √2 ⋅ 1/√2 + (1/√2)²

=  2 + 2 ⋅ 1 +  1/2

=  2 + 2 +  1/2

=  4 + 1/2

=  9/2

4. Answer :

Given : (x + a)(x + b)(x + c)  =  x³ - 10x² + 45 x - 15

x³ + (a+b+c)x² + (ab+bc+ca)x + abc  =  x³ - 10x² + 45x - 15

Compare the coefficients of x² and x.

a + b + c  =  - 10

ab + bc + ca  =  45

We can use the following algebraic identity to find the value of (a² + b² + c²). 

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

or 

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

Subtract 2(ab + bc + ac) from each side. 

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

Substitute.

a² + b² + c²  =  (-10)² - 2(45)

=  100 - 90

=  10

5. Answer :

Given :

2x + 3y  =  13

xy  =  6

To find the value of (8x³ + 27y³), write (8x³ + 27y³) as follows. 

8x³ + 27y³  =  (2x)³ + (3y)³

(2x)³ + (3y)³ is in the form of a³ + b³.

a³ + b³  =  (a + b)³ - 3ab(a + b)

Substitute a = 2x and b = 3y. 

(2x)³ + (3y)³  =  (2x + 3y)³ - 3 ⋅ 2x ⋅ 3y(2x + 3y)

8x³ + 27y³  =  (2x + 3y)³ - 18xy(2x + 3y)

Substitute 2x + 3y  =  13 and xy  =  6. 

=  (13)³ - 18 ⋅ 6 ⋅ (13)

=  2197 - 1404

=  793

6. Answer :

We can write 27x³ + 64y³ as follows. 

27x³ + 64y³  =  (3x)³ + (4y)³

To factor 27x³ + 64y³, we can use the following algebraic identity. 

a³ + b³  =  (a + b)(a² - ab + b²)

Substitute a  =  3x and b  =  4y. 

(3x)³ + (4y)³  =  (3x + 4y)[(3x)² + 3x ⋅ 4y + (4y)²]

27x³ + 64y³  =  (3x + 4y)(9x² + 12xy + 16y²)

7. Answer :

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

8. Answer :

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

9. Answer :

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

10. Answer :

(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

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