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Find x .3*2^2x+2*3^2x =5*6^x.

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for3cast | Student, College Freshman | eNoter

Posted May 19, 2011 at 9:40 AM via web

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Find x .

3*2^2x+2*3^2x =5*6^x.

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giorgiana1976 | College Teacher | Valedictorian

Posted May 19, 2011 at 11:41 AM (Answer #2)

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We notice that 6^x = (2*3)^x

But (2*3)^x = 2^x*3^x

We'll subtract 5*2^x*3^x both sides:

3*2^2x + 2*3^2x - 5*2^x*3^x = 0

We'll divide by 3^2x:

3*(2/3)^2x - 5*(2/3)^x + 2 = 0

We'll note (2/3)^x = t

We'lls square raise both sides and we'll get:

(2/3)^2x = t^2

We'll re-write the equation in t:

3t^2 - 5t + 2 = 0

We'll apply quadratic formula:

t1 = [5+sqrt(25-24)]/6

t1 = (5+1)/6

t1 = 1

t2 = (5-1)/6

t2 = 2/3

Now, we'll put  (2/3)^x = t1:

(2/3)^x = 1

We'll write 1 = (2/3)^0

(2/3)^x = (2/3)^0

Since the bases are matching, we'll apply one to one property:

x = 0

(2/3)^x = 2/3

x = 1

The solutions of the equation are {0 ; 1}

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