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# Value of x

0
157
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+603

Hi friends,

I may have been explained this before, but I just cannot get my head around this, no matter how I try, I just do not grasp this...may I ask just 10 min of your time to just explain this to me again, please..

The sum is: $$M= {{\sqrt{25-2x}} \over x}$$

Determine the value(s) of x for which M will be:

1.1) Real

1.2) Rasional

what confuses me is the WHY

WHY do they equate 25-2x greater and equal to zero for number 1?

for number 2, they find values for x, in this case 8 and 12. They plug them in and get answers...WHY 8 and 12?

ALSO, why not equate 25-2x = 0, or to anything else for number 2?

Rasional numbers are a part of REAL numbers, ....so why not treat them the same?

guys, I really would appreciate your assistance. Thank you all very much..

Oct 11, 2019

#1
+109450
+1

1)    We cannot take the square root of a negative number and get a real number returned

Therfore.....the quantity under the radical MUST BE ≥ 0

So....

25  - 2x ≥  0        add 2x to both sides

25 ≥ 2x              divide both sides by 2

25/2 ≥  x       which is the same as   x ≤ 25/2

Oct 11, 2019
#2
+109450
+1

2)   Any value of x  that makes  25 - 2x  ≥ 0   but  NOT a perfect square  will mean that  a radical will still remain in the expression......and  the remaining radical WILL NOT be a rational number....to see this

Let x  =  2     and we have that

√[25 - (2)(2) ]  / 2  =  √21 / 2       and  √21/2  is NOT RATIONAL

So......we only get a rational  whenever x  = 8   or  x  = 12   thusly

√[25 - 2(8) ] / 2  =  √9 / 2  =   3/2    which  IS rational

And

√ [25 - 2(12) ] / 2  =  √1 /2  =    1/2  which is also rational

So...remember.....any positive integer  under the radical that is not a perfect square will produce an irrational

Hope that helps  !!!

Oct 11, 2019
edited by CPhill  Oct 11, 2019
#3
+603
+1

CPill,

Thank you for your time, I am going to spend some time with examples of these types of questions, however I am sure the light will go on.....I honestly do appreciate!!..have a blessed day...

juriemagic  Oct 11, 2019