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If SQRT(x+15) + SQRT(x)=15 then find the value of x.

+1 vote
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Given x is a real number, If SQRT(x+15) + SQRT(x)=15 then find the value of x.

posted Jan 29, 2017 by anonymous

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8 Answers

+1 vote

It infers sqrt(x+15)=15-sqrt(x) square both sides
x+15=225+x-30*sqrt(x) or
sqrt(x)=7 or x=49 substitute in the statement o.k.

answer Jan 29, 2017 by Kewal Panesar
This solution is simplest, easiest to understand and has fewest steps (though you did leave one step out) therefore the best
Another way to describe Ockham's razor, simplest explanation is probably the answer.
+1 vote

since right side is rational then left side (both parts) must be rational so best method is just substitute in left side to get rational square roots - soon find 49 works

answer May 23, 2017 by anonymous
Actually I solved this both ways.  By adding in numbers high and low (trial and error) I managed to come up with x = 49.

Then I remembered how to square ( a + b) and tried moving squareroot x to other side of equation and it worked.

Sqrt x + 15 = 15 - Sqrt x
(Sqrt x + 15)squared = (15 - Sqrt x)squared
  x + 15.   =  225 - 15Sqrtx - 15Sqrtx + x
  - 210.        =   - 30Sqrtx
      7.       =       Sqrtx
    49.     =      x

This took a few tries for me, as other approaches led to blind
allys algabraicly.

Hope this helps you all.

Robert DiGiovanni
Same approach I took, very fast.
sqrt of 49 + 15 is 8.  8 + 7 = 15    x = 49
0 votes

Given x is a real number, If SQRT(x+15) + SQRT(x)=15 then find the value of x.
ans)
SQRT(x+15)+SQRT(x)=(15)
take square at both side
[SQRT(x+15)+SQRT(x)]^2=(15)^2
(x+15)+2(SQRT(x+15))(SQRT(x))+x=225
2x+15+2SQRT(x+15)x=225
2x+2SQRT(x^2+15x)=225-15
2x+2SQRT(x^2+15x)=210
2SQRT(x^2+15x)=210-2x
now here take square at both side
4(x^2+15x)=(210-2x)^2
4x^2+60x=(44100)-2(210)(2x)+4x^2
now 4x^2 cancle from both side
60x=44100-840x
60x+840x=44100
900x=44100
x=44100/900
x=49
so answer is 49.
thank you..........................................

answer Jan 30, 2017 by Rathod Jigna
this is lengthy
Thank you so much for your explanation.  This 62 year old forgot one of the most basic steps of the FOIL method, thinking I could square the individual terms on the left side of the equation.
0 votes

x=49, SQRT(49+15=64)=8, SQRT(49)=7, 8+7=15

answer Feb 7, 2017 by anonymous
Nice one.
0 votes

I prefer more elegant and fast solutions for puzzles like this:
Let SQRT(x+15)=A and SQRT(x)=B
x should be bigger than 0 and also A>=0 and B>=0 and for sure (A+B)>0

Then

A+B=15+x-x
A+B=A*A - B*B
A+B=(A+B)(A-B)      and A+B>0 =>
1=A-B       (we have also A+B=15)
15+1=(A-B)+(A+B)  => 2*A=16   =>
x+15=64  => x=49

This approach can be used for a lot of similar problems.

answer Mar 1, 2017 by Vasil Arnaudov
sorry I can get to 1=A-B, but lost how U got to 15+1=(A-B)+(A+B)
I lost Vasil at the same place. He's likely thinking faster than I am. So I broke it down into 2 equations this way: 1=A-B or A=B+1, and A+B=15. Substituting then, I get B+1+B=15 or 2B=14 ==> B=7. B=SQRT(x), then x=49
I agree with Vasil, that this is a more elegant (and easier for me) means to a solution.
To get from 1=A-B to 15+1=(A-B)+(A+B), using the fact that A+B=15, Vasil is simply is adding 15 to both sides. Very nice solution.
Solution by inspection: because the sum of two irrationals is also an irrational, x and x + 15 must be perfect squares. By inspection there is only one pair of perfect squares for x and x + 15 whose roots sum to 15, 49 (x) and 64 (x+15), therefore x = 49. QED
If -7 = sqrt(x) shouldn't here be a complex solution as well?
0 votes

It's 49, I promise it's 49

answer Jul 1, 2017 by anonymous
0 votes

Divide the root number (15) by 2, then subtract 1/2 from the result. Square the result and that number is x. 15 divided by 2 = 7.5, Subtract .5 and get 7, square 7 to get 49. 49 is x, works every time with this equation no matter what the root is. x + 15 = 64. sqrt of 64 = 8, sqrt of 49 = 7, 8 + 7 = 15.

answer Jul 28, 2017 by Ed Moody
0 votes

I saw right away that the two terms on the left would each be about half of 15 or about 7.5, so
I guessed x to be 49 and immediately saw 49+15=64 (under the radical) giving 8+7=15 which checked out. The lesson here, I think, is the importance of approximating an answer before diving into the algebra.

answer Aug 2, 2017 by anonymous
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