Question

if x=1/4-x, find : x+1/x

Answer 1

Answer:

x  + 1/x  = 4

x³ + 1/x³  = 52

x⁶ + 1/x⁶ = 2702

Step-by-step explanation:

x = 1/(4 -x)

=> x (4 -x) = 1

=> 4x - x² = 1

=> x² - 4x + 1 = 0

=> x² + 1 = 4x

Dividing by x both sides

=> x + 1/x = 4

or solving Quadratic equation

x² - 4x + 1 = 0

=> x = (4 ± √16 - 4 )/2

= (4 ± 2√3)/2

= 2 ± √3

case 1  x  = 2 + √3

x + 1/x

= 2 + √3  +  1/(2 + √3)

Multiplying & dividing last term by 2 - √3

= 2 + √3    + (2 - √3)/(4 -3)

= 2 + √3 + 2  - √3

= 4

Case 2 x  = 2 - √3

x + 1/x

= 2 - √3  +  1/(2 - √3)

Multiplying & dividing last term by 2 + √3

= 2 - √3    + (2 + √3)/(4 -3)

= 2 - √3 + 2  + √3

= 4

x  + 1/x  = 4

x³ + 1/x³ = (x + 1/x)³ - 3x(1/x)(x + 1/x)

= 4³ - 3 * 4

= 64 - 12

= 52

x²+ 1/x² = (x + 1/x)² - 2x(1/x)

=> x²+ 1/x² = 4² - 2

=> x²+ 1/x² = 14

Cubing both sides

x⁶ + 1/x⁶ + 3x²(1/x²)(x²+ 1/x²) = 14³

=> x⁶ + 1/x⁶ + 3(14) = 2744

=> x⁶ + 1/x⁶ = 2702

Answer 2

Answer:

4

Step-by-step explanation:

i.x=1/4-x

or 4x-x^2=1

or 4-x=1/x (diving both sides by x)

or 4=x+1/x

ii. (x+1/x)^3=x^3+1/x^3+3(x+1/x)

or 64= x^3+1/x^3+12

x^3+1/x^3= 52

iii. (x^3+1/x^3)^2=x^6+1/x^6+2

x^6+1/x^6=2702