Algebraic Identities With solved examples

Algebraic Identitites by Gourav Tomar

Question based on algebraic identities

Ex.1 x² + 1/x² = 7 Then find the Value of x³ + 1/x³.

Sol. (x + 1//x)² = x² + 1/x² + 2*x*1/x = 7+2 = 9

(x + 1/x) = √9 = 3

Cubing both sides

x + 1/x = 3³

x³ + 1/x³ + 3*3  = 27 

x³ + 1/x³ = 27-9 = 18

Shortcut method using formula

x² + 1/x² =7

x + 1/x = √(7+2) = 3

x³+ 1/x³ = 3³ - 3*3 = 27-9= 18 Ans.

Ex.2 If x = 3 + 2√2 then the value of √x - 1/√x ?

Sol. x = 3+ 2 √2 

Rationalize the equation

1/x = 1/ (3+2√2) = 1/(3+2√2) * (3-2√2)/(3-2√2) = (3-2√2) /(9-8) = 3-2√2

(√x- 1/√x)² = x +(1/x) - 2

Insert value of  x in above equation

3 + 2√2 + 3 - 2√2 -2 = 4

4 =(√x- 1/√x)²

√x - 1/x = 2  Ans

Ex.3 If m + 1/ (m-2) = 4 Then find the value of (m-2)² + 1(m-2)² .

Sol. m + 1/(m-2) = 4

(m-2) + 1/(m-2) = 2 

Now, 

(m-2)² + 1/(m-2)²  = [(m-2)²+ 1/(m-2)²]-2 

2²-2 = 2 Ans

Ex.4 If x= 4ab/a+b then find the value of (x+2a)/(x-2a) + (x+2b)/(x-2b).

Sol. x = 4ab/(a+b) 

 x/2a = 2b/a+b

By componendo and Dividendo

(x+2a)/(x-2a) = (3b+a)/b-a

Similarily,

(x+2b)/(x-2b) = (3a+b)/(a-b)

(x+2a)/(x-2a) + (x+2b)/(x-2b) = (3b+a)/(b-a) + 3a+b)/(a-b)

(3b+a-3a-b)/b-a = (2b-2a)/b-a

2(b-a)/(b-a) = 2 Ans

Ex.5 If x = 2- 2^1/3 +2^2/3 Then find the value of x³- 6x² + 18x .

Sol. x= 2- 2^1/3 + 2^2/3

x-2 = 2^1/3 + 2^2/3.

On cubing both the sides

x³ - 8 - 6x² + 12x = 4 -2 -6 (x-2).

x³ - 6x² + 12x + 6x = 4-2+12+8

x³- 6x² + 18x = 22 Ans

Ex.6 a² = b+c, b² = a+c, and c²= a+b then find the Value of 1/(1+a) + 1/(1+b) + 1/(1+c),.

Sol. 1/(1+a) + 1/(1+b) + 1/(1+c)

Multiply and divide a/a ,b/b, c/c respectively in the above equation.

a/(a+a²) + b/(b+b²) + c/(c+c²)

Put values of a², b² , c² in the above equation

a/(a+b+c) + b/(a+b+c) + c/(a+b+c)

(a+b+c)/(a+b+c) = 1 Ans

Ex.7 If a/(1-a)+ b/(1-b) + 1/(1-c) = 1 Then find the value of 1/(1-a) +1/(1-b) + 1/(1-c) .

Sol. a/(1-a)+ b/(1-b) + 1/(1-c) = 1

Add 1 in each term in the above equation

[a/(1-a) + 1] + [1/(1-b) + 1] + [1/(1-c) + 1] = 1+3

(a+1-a)/(1-a) + (b+1-b)/(1-b) + (c+1-c)/(1-c) = 4

1/(1-a) +1/(1-b) + 1/(1-c) = 4  Ans.

Ex.8 If a, b, c are real and a² + b² + c² = 2(a-b-c)-3, then value of 2a-3b+4c ?

Sol. a²+b²+c² = 2a-2b-2c-3

a²-2a+b²+2b+c²+2c+1+1+1=0

(a²-2a+1) + (b²+2b+1) + (c²+2c+1) = 0

(a-1)² + (b+1)² + (c+1)².

a-1 = 0 , b-1=0 , c-1= 0

a=1, b=-1, c=-1 

2a-3b+4c = 2+3-4 = 1 Ans

Ex.9 a/b = 25/6, then the value of (a²-b²)/(a²+b²) .

Sol. a²/b² = 25²/6² = 625/36

By componendo and dividendo,

(a²-b²)/(a²+b²) = (625-36)/(625+36) = 589/661  Ans.

Ex.10 If x+y+z = 6 and x²+y²+z² = 20 then the value of x³+y³+z³-3xyz is?

Sol. x+y+z = 6

On squaring,

x²+y²+z²+2xy+2yz+2zx = 36

20+2(xy+yz+zx) = 36

xy+yz+zx = 8

x³+y³+z³-3xyz = (x+y+z)(x²+y²+z²-xy-yz-zx)

6(20-8)

72 Ans

Ex.11 If x + 1/x = 2 then x²⁰¹³ + 1/x²⁰¹⁴?

Sol. x + 1/x = 2

x2 + 1 = 2x

x²-2x+1 = 0

(x-1)²= 0

x = 1

x²⁰¹³ + 1/x²⁰¹⁴  = 1+1 = 2 Ans