SOLUTION: 7-sqrt(1-4x)-sqrt(4-6x)=0 I have no idea how to do this problem. Please help!

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Question 836280: 7-sqrt(1-4x)-sqrt(4-6x)=0
I have no idea how to do this problem. Please help!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
This is a long and messy problem. The basic idea is this though: Isolate the square roots one by one on their own side. That will allow you to square both sides to eliminate each square root one by one. After doing all that, you can use the quadratic formula to solve for x. Remember to check ALL possible answers.

7-sqrt%281-4x%29-sqrt%284-6x%29=0

7-sqrt%281-4x%29=sqrt%284-6x%29

%287-sqrt%281-4x%29%29%5E2=%28sqrt%284-6x%29%29%5E2

%287-sqrt%281-4x%29%29%287-sqrt%281-4x%29%29=4-6x

7%287-sqrt%281-4x%29%29-sqrt%281-4x%29%287-sqrt%281-4x%29%29=4-6x



49-7sqrt%281-4x%29-7sqrt%281-4x%29%2B%28sqrt%281-4x%29%29%5E2=4-6x

49-7sqrt%281-4x%29-7sqrt%281-4x%29%2B1-4x=4-6x

50-4x-14sqrt%281-4x%29=4-6x

-14sqrt%281-4x%29=4-6x-50%2B4x

-14sqrt%281-4x%29=-46-2x

%28-14sqrt%281-4x%29%29%5E2=%28-46-2x%29%5E2

196%281-4x%29=%28-46-2x%29%28-46-2x%29

196%281-4x%29=-46%28-46-2x%29-2x%28-46-2x%29

196%281%29%2B196%28-4x%29=-46%28-46%29-46%28-2x%29-2x%28-46%29-2x%28-2x%29

196-784x=2116%2B92x%2B92x%2B4x%5E2

0=2116%2B92x%2B92x%2B4x%5E2-196%2B784x

0=4x%5E2%2B968x%2B1920

4x%5E2%2B968x%2B1920=0

4%28x%5E2%2B242x%2B480%29=0

x%5E2%2B242x%2B480=0%2F4

x%5E2%2B242x%2B480=0

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Now use the quadratic formula to solve for x

Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve x%5E2%2B242%2Ax%2B480=0 ( notice a=1, b=242, and c=480)





x+=+%28-242+%2B-+sqrt%28+%28242%29%5E2-4%2A1%2A480+%29%29%2F%282%2A1%29 Plug in a=1, b=242, and c=480




x+=+%28-242+%2B-+sqrt%28+58564-4%2A1%2A480+%29%29%2F%282%2A1%29 Square 242 to get 58564




x+=+%28-242+%2B-+sqrt%28+58564%2B-1920+%29%29%2F%282%2A1%29 Multiply -4%2A480%2A1 to get -1920




x+=+%28-242+%2B-+sqrt%28+56644+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-242+%2B-+238%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-242+%2B-+238%29%2F2 Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


x+=+%28-242+%2B+238%29%2F2 or x+=+%28-242+-+238%29%2F2


Lets look at the first part:


x=%28-242+%2B+238%29%2F2


x=-4%2F2 Add the terms in the numerator

x=-2 Divide


So one answer is

x=-2




Now lets look at the second part:


x=%28-242+-+238%29%2F2


x=-480%2F2 Subtract the terms in the numerator

x=-240 Divide


So another answer is

x=-240


So our solutions are:

x=-2 or x=-240





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So the *possible* solutions are x+=+-2 or x+=+-240

However, we must check them both

Checking x+=+-2

7-sqrt%281-4x%29-sqrt%284-6x%29=0

7-sqrt%281-4%28-2%29%29-sqrt%284-6%28-2%29%29=0

7-sqrt%281%2B8%29-sqrt%284%2B12%29=0

7-sqrt%289%29-sqrt%2816%29=0

7-3-4=0

0=0 This is TRUE, so x+=+-2 is definitely a solution.

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Checking x+=+-240

7-sqrt%281-4x%29-sqrt%284-6x%29=0

7-sqrt%281-4%28-240%29%29-sqrt%284-6%28-240%29%29=0

7-sqrt%281%2B960%29-sqrt%284%2B1440%29=0

7-sqrt%28961%29-sqrt%281444%29=0

7-31-38=0

-62=0 This is FALSE, so x+=+-240 is NOT a solution.


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Final Answer: x+=+-2