Question

Find all values of n,[where n is a whole number]for which the equation (x-8)/(n-10)=(n)/(x) has no solutions for x

Answer 1

Given :  Equation (x-8)/(n-10)=(n)/(x) has no solutions for x  

To find :  all values of n  where n is a whole number  

Solution:

(x-8)/(n-10)=(n)/(x)        n - 10 is in denominator hence n can not be 10

=> x(x - 8) = n(n - 10)

=> x² - 8x  = n² - 10n

=> x²  - 8x   + 10n - n² = 0

does not have any  solution

iff D < 0

iff  8² - 4(10n - n²) < 0

iff 4n² - 40n  + 64 < 0

iff n² - 10n + 16  < 0

iff (n - 2)(n - 8)  < 0

=>   2 < n < 8

n = 3  , 4 , 5 , 6 , 7    

for n = 3  , 4 , 5 , 6 , 7  , 10   has no solution for x  

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