Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

find x

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: find x

x=2 is one solution, and there's another somewhere in (-3,-2).

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LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: find x

mathsy,

what is the method to find such solutions?

TI-89 says the two solutions are:

x = 2
x = -ln(250)/ln(10)

Last edited by LuisRodg (2008-11-11 09:01:46)

Prakash Panneer

Member

Registered: 2006-06-01

Posts: 110

Re: find x

Given that

2^(x+2) 5^(6-x) = 10^(x^2)
                         = (2 x 5)^(x^2)
                         =2^(x^2) 5^(x^2)
Compare the LHS and RHS

x + 2 = x^2         and             6 - x = x^2
x^2 - x - 2 = 0                         x^2 + x - 6 = 0

x = 2, -1   and x = 2, -3

This is our required answer.

Last edited by Prakash Panneer (2008-11-11 09:34:45)

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Letter, number, arts and science
of living kinds, both are the eyes.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: find x

On the integers it's easy to see that the RHS of the equation is a power of 10, which means you need to have the same number of 2's and 5's on the LHS.  It just so happens that the value of x at which this occurs makes the entire equation true.

Non-integer solutions are more tricky.  You need to do some fancy footwork with logarithms:

Now you can use the quadratic formula to solve for x.

The + half turns out to be x = 2.  Whether or not the - half matches your TI-89's answer is beyond me, as I'm not sure how to simplify the expression further.

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Prakash Panneer

Member

Registered: 2006-06-01

Posts: 110

Re: find x

What my answer is right only.
for example...

a^3 = a^x

We can take x = 3, since the bases are equal.

Similarly we can solve the given problem.

Hence the answer would be...

2^(x+2) 5^(6-x) = 10^(x^2)
                         = (2 x 5)^(x^2)
                         =2^(x^2) 5^(x^2)

Compare LHS and RHS

x + 2 = x^2         and             6 - x = x^2
x^2 - x - 2 = 0                         x^2 + x - 6 = 0

x = 2, -1   and x = 2, -3

This is our required answer.

Last edited by Prakash Panneer (2008-11-11 10:37:11)

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Letter, number, arts and science
of living kinds, both are the eyes.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: find x

The + half turns out to be x = 2.  Whether or not the - half matches your TI-89's answer is beyond me, as I'm not sure how to simplify the expression further.

As long as you're convinced that the + half is 2, you can use that fact to express the - half more simply.

Not sure if that helps, but it's better than before.

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Why did the vector cross the road?
It wanted to be normal.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: find x

Yep, it's easy to pull LuisRodg's second answer from that:

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