Can you solve it
If x > 1 satisfy x + x 1 = 4 , find x − x 1 .
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3 solutions
(x+ x 1 )=4
(x+ x 1 )²=4² ;(By square on both sides)
x²+2.x. x 1 + x ² 1 =16
x²+2+ x ² 1 =16
x²+ x ² 1 =16-2=14
x²+ x ² 1 =12+2
x²-2+ x ² 1 =12
x²-2.x. x 1 + x ² 1 =12
(x- x 1 )²=12 ;{(a-b)²=a²-2ab+b²}
(x- x 1 )²= (2√3)² ;{(2√3)²=2².(√3)²=4.3=12}
x- x 1 =2√3 ;(by square root on both sides)
(ans.)
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Whats different in my solutio bro???
Notice that at the end, when you take the square root of both sides, x − x 1 = ± 2 3 . How do you know the positive root is the correct one? (Hint: you haven't used the fact that x > 1 yet)
x + 1/x = 4, x^2 - 4x + 1 =0, 2x = 4 + sqrt(16 - 4), x = 2 + sqrt(3), x - 1/x = 2+ sqrt(3) - 1/(2 + sqrt(3)) = 2 + sqrt(3) - (2 - sqrt(3) = 2*sqrt(3).
( a + b ) 2 = ( a − b ) 2 + 4 a b
Put a = x , b = x 1
( x + x 1 ) 2 = 4 + ( x − x 1 ) 2
⇒ ( x − x 1 ) 2 = 1 2
⇒ x − x 1 = 2 3 We neglect − 2 3 as it doesn’t satisfy the given expression.
@Munem Sahariar The condition of x > 1 is extraneous ..so please delete that part.
x + x 1 = 4
So x 2 + x 2 1 = 1 4
Hence x 2 + x 2 1 − 2 = 1 2
So ( x − x 1 ) 2 = ( 2 3 ) 2
Hence x − x 1 = 2 3