Importance of 10

Algebra Level 4

Let f ( x ) = ( 1 − a ) x 2 + ( 1 + a b − b ) x + 1 + b 2 , \displaystyle f(x) = (1 - a) x^2 + (1 + ab - b) x + 1 + b^2, where a a and b b are single-digit natural numbers. Given that one of the roots of f ( x ) f(x) is 10 , 10, find the other root.


The answer is -2.5.

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Rajen Kapur by Rajen Kapur , 72, India

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1 solution

Rajen Kapur
Jul 7, 2015

Given f ( x ) = 0 f(x) = 0 on transposing and factorizing R.H.S. x 2 + x + 1 = a x 2 + ( b − a b ) x − b 2 = ( x − b ) ( a x + b ) \displaystyle x^2 + x + 1 = a x^2 + (b - ab) x - b^2 = (x - b) ( a x + b) which at x = 10 x = 10 is, L.H.S. 100 + 10 + 1 = 111 = (3) * (37) = (10 - 7) (30 + 7) equalling R.H.S. ( 10 - b) (10 a + b). It is patently clear that a = 3 and b= 7 (given that a and b are single-digit natural numbers.) Using a and b in f ( x ) = 0 f(x) = 0 , − 2 x 2 + 15 x + 50 = ( 10 − x ) ( 2 x + 5 ) = 0 \displaystyle -2 x^2 + 15 x + 50 = (10 - x) (2 x + 5) = 0 gives the answer -2.5

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Sometimes it's necessary to take a far away look of the problem. Too much close makes us blind. Fine Solution!!!

Cleres Cupertino - 5 years, 10 months ago

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