SOLUTION: 2^2x+3 -9(2^x)+1=0

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Question 422305: 2^2x+3 -9(2^x)+1=0
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!

The easiest way I see to solve this problem is not especially easy. It requires a good understanding of exponents and quadratic equations. So we will start with a short review of a couple of important ideas:
So to solve your equation we will rewrite the first term, using in reverse, so that its exponent is twice the exponent of your middle term:

And since this becomes:

or

We now have the equation in quadratic form. The first few times yousolve one of these, it can be helpful to use a temporary variable. Set it equal to the base and exponent of the middle term:
Let
Then
Substituting these into our equation we get:

This is clearly a quadratic equation. To solve it we factor it of use the Quadratic Formula. This factors fairly easily:
(8q-1)(q-1) = 0
From the Zero Product Property we know that one of these factors must be zero. So:
8q-1 = 0 or q-1 = 0
Solving these we get:
q = 1/8 or q = 1
Of course we are not interested in solutions for q. We are interested in solutions for x. So now we substitute back in for q. (Remember, q was just a temporary variable).
or
Solving these for x we get:
x = -3 or x = 0
These are two solutions to your equation. Check them if you like.

Note: Once you have done a few of these quadratic form equations you will no longer need a temporary variable. You will see how to go directly from

to

to
or
etc.

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