Question

if x=13/√19+8√3 find the exact value of x^4-6x^3-2x^2+18x+23/x^2-8x+15

Answer 1

Given :  x = √(19 - 8√3)  

To find :  ( x⁴  - 6x³  - 2x²  + 18x  + 23  )/ (x² - 8x  + 15 )

Solution:

x = √(19 - 8√3)   = √(16 + 3  - 8√3)  = √(4² + (√3)² - 2(4)√3))

=> x =  √( 4 - √3)²

=> x = 4 - √3

 x² = 19 - 8√3

 x⁴  - 6x³  - 2x²  + 18x  + 23

= x²(x² - 2)  - 6x(x² - 3)  + 23

= (19 - 8√3 )(19 - 8√3 - 2)   - 6(4 - √3)(19 - 8√3 - 3)   + 23

= (19 - 8√3 )(17 - 8√3)   - 6(4 - √3)(16 - 8√3)   + 23

= 323 + 192 - 288√3    - 6(64 + 24 - 48√3)  + 23

= 323 + 192 - 288√3    -528 +  288√3  + 23

= 538 - 528

= 10

x² - 8x  + 15  

=  19 - 8√3   - 8(4 - √3)  + 15

=  19 - 8√3   - 32 + 8√3   + 15

= 2

( x⁴  - 6x³  - 2x²  + 18x  + 23  )/ (x² - 8x  + 15 )

= 10/2

= 5

( x⁴  - 6x³  - 2x²  + 18x  + 23  )/ (x² - 8x  + 15 )  = 5

Learn more:

Solve the following x+y/xy=5 and x-5/xy=7 - Brainly.in

https://brainly.in/question/8168066

4^x + 6^x = 9^x. Find the value of x

https://brainly.in/question/13718805