9. If x^2 + 4x -1 = 0 and x is positive then find the value of
(a) x + 1 (b) x² + 1/x^2.
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Correct Question:
If x² + 4x - 1 = 0 , and x is positive then find the value of ;
a). x + 1/x
b). x² + 1/x²
Answer:
a). x + 1/x = 2√5
b). x² + 1/x² = 18
Solution:
We have ;
=> x² + 4x - 1 = 0
=> x² + 4x + 2² - 2² - 1 = 0
=> x² + 2•x•2 + 2² = 2² + 1
=> (x + 2)² = 5
=> x + 2 = √5
=> x = √5 - 2 ( x > 0 )
Now,
=> x = √5 - 2
=> 1/x = 1/(√5 - 2)
=> 1/x = (√5 + 2)/(√5 - 2)(√5 + 2)
=> 1/x = (√5 + 2)/[ (√5)² - 2² ]
=> 1/x = (√5 + 2)/(5 - 4)
=> 1/x = (√5 + 2)/1
=> 1/x = √5 + 2
Now,
=> x + 1/x = √5 - 2 + √5 + 2
=> x + 1/x = 2√5
Hence,
x + 1/x = 2√5
Now,
=> x + 1/x = 2√5
=> (x + 1/x)² = (2√5)²
=> x² + 2•x•(1/x) + (1/x)² = 20
=> x² + 2 + 1/x² = 20
=> x² + 1/x² = 20 - 2
=> x² + 1/x² = 18
Hence,
x² + 1/x² = 18
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