Question 426593: Need help solving this equation with rational exponents please.
Solve equation. Check answer.
x^(3/4) - 2x^(1/2) - 4x^(1/4) + 8 = 0 Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
The key to solving this is to see that the the exponents on x are "consecutive" in a sense. If you look at 1/2 as 2/4, you can see that the exponents are 3/4, 2/4 and 1/4. Because of this pattern we can solve this equation using techniques similar to those we use on "regular" polynomials.
Using a temporary variable can help make this easier:
Let
then
and
Now we substitute these into your equation:
This now looks like a "regular" polynomial. To solve we will factor it. It will factor by grouping:
The second factor is a difference of squares so we can factor it further:
From the Zero Product Property we know that one of these factors must be zero. So:
q-2 = 0 or q+2 = 0
Solving these we get:
q = 2 or q = -2
These are solutions for q. But we're not interested in solutions for q. We are interested in solutions for x. So now we substitute back in for q: or
Since an exponent of 1/4 means 4th root and since even-numbered roots are positive, there is no way for the second equation to be true. So our solution(s) will only come from the first equation. Raising both sides of the first equation to the 4th power we get:
Because we raised both sides to an even power we must check our answer. We use the orginal equation to check:
Checking x = 16:
which simplifies as follows:
8 -2(4) -4(2) + 8 = 0
8 - 8 - 8 + 8 = 0
0 = 0 Check!!
So the only solution to your equation is x = 16.
P.S. After a few of these types of problems you will no longer need the temporary variable. You will see how to go from
to
to
to
etc.
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