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One way to solve this would be to start be rewriting the equation in terms of . We can use the division rule for exponents, in reverse, on the second term to get:
The numerator simplifies to 16:
Now we can simplify the equation my eliminating the fraction. This can be done by multiplying both sides of the equation by :
which simplifies to:
Next we get a zero on the right side by subtracting that term from each side:
Since the exponent of is twice the exponent of the exponent in the middle term, , this equation is in what is called "quadratic form". These kinds of equations can be solved just like "regular" quadratic equations. To make it easier to understand I am going to use a temporary variable:
Let
The
Substituting these into the equation we get:
This obviously a quadratic equation and factoring it is easy:
By the Zero Product Property:
q-1 = 0 or q-16 = 0
Solving these we get:
q = 1 or q = 16
We have solved for q. But we want to solve for x. Now we substitute for the temporary variable: or
Only one power of 2 is equal to 1 and that is zero. So x = 0 is the only solution for the first equation. And there is only one power of 2 that is 16 and that is 4. So the only solution to the second equation is x = 4. Putting these together the complete solution to your equation is:
x = 0 or x = 4
Once you have done a few "quadratic form" problems like this you will no longer need the temporary variable. You will see how to go from:
to
to or
etc.
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