Question

Determine in the following case whether there exists a real number x satisfying the indicated relation, and if there is, determine this number.

Answer 1

The answer key gives says there is no real number x, but I am having trouble understanding this.

\[\sqrt{x-2} = 3 - 2 \sqrt{x}\]

Answer 2

So what I've tried:

\[x - 2 = 9 - 12\sqrt{x} + 4x\]

Answer 3

then:
\[-2 = 9 - 12\sqrt{x} + 3x\]
\[-11 = - 12\sqrt{x} + 3x\]

Answer 4

And then I guess I could divide by - 12? Which gives:

\[\frac{ 11 }{ 12 } = \sqrt{x} - \frac{ 1 }{ 4 } x\]

But that doesn't really seem to help.

Answer 5

subtract 3x from both sides of this equation

\[\Large -11 = - 12\sqrt{x} + 3x\]

then square both sides

Answer 6

So
\[-11-3x = -12\sqrt{x}\]
\[121 + 66x + 9x^2= 144x\]

Answer 7

now get everything to one side and then use the quadratic formula

Answer 8

So you should get a rational number because you can get a number with the quad. formula?

Answer 9

actually nope. because number is neg in square. Thank you!

Answer 10

you should get real number solutions for that last equation you got

Answer 11

`The answer key gives says there is no real number x`

that's not correct. There is one real number solution

Answer 12

Yeah. I'm getting:
\[\frac{ 88\pm \sqrt{3388} }{ 18 }\]

Answer 13

neither of those are what I'm getting

Answer 14

Oh. I did some bad arithmetic. Let's see.

\[9x^2 -78x + 121 = 0\]

Answer 15

so far so good

Answer 16

So \[\frac{ -78 \pm \sqrt{1728} }{ 18 }\]

Answer 17

nope

Answer 18

That should be a positive 78

Answer 19

good

Answer 20

now you must check each potential solution back into the original equation

Answer 21

what I would do is find the decimal form of each solution, and then plug them in to test

Answer 22

example: x = 1/2 is a bit clunky but x = 0.5 is easier to work with

Answer 23

Okay, so the answer is
\[\frac{ 78 - \sqrt{1728} }{ 18 }\]

Answer 24

Thank you!

Answer 25

correct

Answer 26

you're welcome