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

Question

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

Answer 1

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.

Comments

This solution is simplest, easiest to understand and has fewest steps (though you did leave one step out) therefore the best

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Another way to describe Ockham's razor, simplest explanation is probably the answer.

Answer 2

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

Comments

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

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Same approach I took, very fast.

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sqrt of 49 + 15 is 8.  8 + 7 = 15    x = 49

Answer 3

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..........................................

Comments

this is lengthy

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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.

Answer 4

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

Comments

Nice one.

Answer 5

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.

Comments

sorry I can get to 1=A-B, but lost how U got to 15+1=(A-B)+(A+B)

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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.

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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.

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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

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If -7 = sqrt(x) shouldn't here be a complex solution as well?

Answer 6

It's 49, I promise it's 49

Answer 7

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 8

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.