The x sits on top!
1 2 5 x + 4 5 x = 2 × 2 7 x
Find the number of real values of x that satisfy the above equation.
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The answer is 1.
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2 solutions
Nice solution. I divided through by 3 3 x to get the equation
( 3 5 ) 3 x + ( 3 5 ) x − 2 = 0 ,
which is the form y 3 + y − 2 = ( y − 1 ) ( y 2 + y + 2 ) = 0 ,
where y = ( 3 5 ) x > 0 , so y = 1 ⟹ x = 0 is the only soution.
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Did the same way ! :p
i think essentially we have done the same thing,right?
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Pretty much. I just thought my slightly different presentation was worth a quick mention.
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@Brian Charlesworth – yes,definitely it is!
and i feel your solution is somewhat more complete than mine as i simply state that 0 is the only solution leaving the reasoning to the reader whereas u completely reason it out
Okay, x=0 satisfies but the answer is 1.-.
Or intutively, on dividing by 27^x both sides, we get a increasing function on LHS and a constant function on RHS, this now goes by intution and logic that the equatiom can have atmost 1 real root. (x=0)
Relevant wiki: Factorization of Polynomials
The equation can be rewritten as 5 3 x − 3 3 x + 3 2 x ( 5 x − 3 x ) = 0
( 5 x − 3 x ) ( 5 2 x + 2 . 3 2 x + 1 5 x ) = 0
Clearly,only x = 0 satisfies the given condition