How Many Solutions Exist!

Algebra Level 3

How many solutions are there for the following expression?

( 3 x ) 2 + x + ( 4 x ) 2 + x − ( 6 x ) 2 + x = 1 (3^x)^{2+x} + (4^x)^{2+x} - (6^x)^{2+x} =1

1 0 3 4 2

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Let 3 x ( 2 + x ) = a 3^{x(2+x)}=a and 2 x ( 2 + x ) = b 2^{x(2+x)}=b . Then the given equation can be written as a + b 2 − a b = 1 a+b^2-ab=1 , or ( 1 − b ) ( a − b − 1 ) = 0 (1-b)(a-b-1)=0 . Therefore 1 − b = 0 1-b=0 or b = 1 b=1 , which yields x = 0 x=0 or x = − 2 x=-2 . Or a = b + 1 a=b+1 , which yields x = − 1 + 2 x=-1+\sqrt {2} or x = − 1 − 2 x=-1-\sqrt {2} . So, there are 4 \boxed {4} solutions in all.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

NIce solution. Thank you.

Hana Wehbi - 1 year, 7 months ago

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