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Question 595477: What is the solution to the equation to the square root of 2X+1 minus the fourth root of X + 11=0
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! To start with, please use parentheses, to group multiple terms which should be treated as a unit, even if your post is in words. This includes but is not limited to:- The radicand of any kind of root, including square roots. ("Radicand" is the name for the expression inside a radical.)
- Numerators
- Denominators
- Exponents
Without parentheses your expressions are not clear. And tutors are less likely to help when the problem is unclear. (This may be why it has taken this long for a tutor to respond.)
Your problem without the parentheses means:
which I suspect is not right.
I'm going to guess that the problem is really:
If my guess is wrong, then a) please re-post using parentheses; b) keeping reading (you might learn something).
Solving equations where the variable is in a radicand usually involves raising each side of the equation to some power so that the radical "goes away". For this to work, one radical must be isolated on one side of the equation. So we will start by adding the 4th root to each side. This gives us:
Now, as it happens, both radicals are isolated. Now we just have to figure out what power to use. We want to use a power that will make at least one radical disappear. If the radicals were both the same kind of root, then the power to use would be obvious.
With different kinds of roots, choosing the power is more difficult. To figure out the power it can help to rewrite the radicals as fractional exponents. This makes our equation:
The best power to use is the lowest common denominator (LCD) of the denominators in the exponents. The LCD of 2 and 4 is 4. So 4 is the best power to use.
To simplify we start by using the exponent rule for a power of a power: Multiply the exponents.
which simplifies to:
And the radicals (and fractional exponents) are gone. (Note: Using fractional exponents is not required if you can see all of this without them.)
Now we have an equation we can solve. Start by simplifying each side. The right side is easy. Be careful when squaring the right side. Exponents do not distribute. To square it correctly you must use FOIL on (2x + 1)(2x + 1) or use the  pattern with the "a" being 3x and the "b" being 1. I like using the pattern:
which simplifies to:
This is a quadratic equation. So we want one side to be zero. Subtracting x and 11 from each side we get:
Now we factor (or use the Quadratic Formula). The possible factors for  are either 4x and x or 2x and 2x. For -10 the possible factors are -1 and 10, 1 and -10, -2 and 5 or 2 and -5. With so many factors it could take quits a while to find the right combination or, worse, to find out that the expression will not factor at all. So don't feel bad if you decide that the Quadratic Formula would be easier.
Here's some tips on how to factor expressions like this more easily:- Use the discriminant (from the Quadratic Formula),
 , to determine quickly if the expression will factor. If the discriminant is not a perfect square then- the expression will not factor and the Quadratic Formula must be used; and
- you already have the value for the discriminant for the formula
Your discriminant:
Since 169 is a perfect square. we know that the equation will factor. - Once you know that factoring will work try to eliminate some of the combinations before you start mechanically plowing your way through them. Sometimes some combinations can be eliminated from contention.
"b", the coefficient of the middle term, is the key. As you probably know the middle term is the result of adding two like terms. Let's see how this helps us eliminate some of the possible combinations:
Your "b" is 3 which is an odd number. You may have not realized this before but the only way to get an odd when adding two numbers is to add an even and an odd. So one of the products must be odd. And, another fact you may not have realized, the only way to get an odd when multiplying two numbers is to multiply two odds!
This tells us that a combination using the 2x and 2x factors of cannot work since each product will have a 2x in them!
Also, the only odd factor of 4x and x, the 1 from 1x, must be multiplied by an odd factor of -10 in order to get that odd product we need.
So just from the fact that "b" was odd we have already eliminated 8 of the 12 possible combinations. The only combinations that could work are:
(4x + 1)(1x - 10)
(4x - 1)(1x + 10)
(4x - 5)(1x + 2)
(4x + 5)(1x - 2)
It should not take a long time to figure out the right one. (The third one is the only one that would result in "b" being +3.)
So the factored equation is
(4x - 5)(1x + 2) = 0
From the Zero Product Property we that this (or any) product can be zero only if one of the factors is zero. So:
4x - 5 = 0 or x + 2 = 0
Solving these we get:
x = 5/4 or x = -2
Last of all the must check our answers. This is required whenever you solve equations where the variable is in one or more even-numbered roots. Even if no mistakes have been made it is possible that one or more of your solutions to not actually work in the original equations. These "don't work" solutions are called extraneous solutions. Extraneous solutions must be rejected and they can only be detected by checking.
Use the original equation to check.
Checking x = 5/4
Simplifying:
Rationalizing:
It doesn't appear that this answer is checking out. But we can use fractional exponents to help us simplify the 4th root:
And presto! x = 5/4 checks!
Checking x = -2
Simplifying:
We can see that the radicand of the square root is negative. This is not allowed. So x = -2 is extraneous and must be rejected.
So the only solution to
is
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