The solution set for the equation √x+14 - √2x+5 = 1

Question

How to solve for X?

The solution set for the equation √x+14 - √2x+5 = 1

Answer 1

Hi there! Here is a step-by-step solution :)

 Isolate

     √x+14 = √2x-5+1

Raise both sides to the second power

     (√x+14)² = (√2x-5+1)²

After squaring

     x+14 = 2x-5 + 1 + 2√2x-5

Current equation

     x+14 = 2x-5 + 1 + 2√2x-5

     Isolate radical on the left hand side

     -2√2x-5 = -x-14 + 2x-5 +1

     Tidy up

     2√2x-5 = -x+18

 Raise both sides to the second power

     (2√2x-5)² = (-x+18)²

     After squaring

     8x-20 = x²-36x + 324

Rearranged equation

     x²  - 44x  + 344 = 0

     This equation has two real roots:

      x₁ = ( 44 + sqrt(560) ) / 2 = 33.8322

     x₂ = ( 44 - sqrt(560) ) / 2 = 10.1678

Check that the first solution is correct:

     Original equation, root isolated, after tidy up

     √x+14 = √2x-5+1

     Plug in 33.8322 for  x 

      √(33.8322) +14 = √2•(33.8322) -5+1

      Simplify

      √47.832 = 8.916

      Solution does not check

      6.916 ≠ 8.916 

Check that the second solution is correct:

  Original equation, root isolated, after tidy up

     √x+14 = √2x-5+1

     Plug in 10.1678 for  x 

      √(10.1678) +14 = √2•(10.1678) -5+1

      Simplify

      √24.168 = 4.916

Solution checks!

4.916 = 4.916

Solution is:

      x = 10.1678

Let me know if you have any questions!

Answer 2

x= about 1,684.4, rounded to 1 decimal

if you meant (sqrx) +14 - (sqr2x) + 5 = 1

and not sqr(x+14) - sqr(2x+5) = 1

if the latter, then x=2 is the solution

it came out so evenly, that's probably what you want

but as written, your problem is the former with the 1,684.4 solution

there's a 3rd way to interpret the problem that also is probably not what you meant as it also gives a "sloppy" decimal answer. (sqrx) +14 - (sqr2)x +5 = 1 Just unlikely you meant that but possible

sometimes half the problem is figuring out what the problem really is

jump to the end of this answer for the answer you likely want

sqrx + 14 - sqr2x +5 =1

sqrx = 1-14 -5 + sqr2x

sqrx = sqr2x - 18

square both sides

x = 2x -36sqr2x + 18^2

combine like terms

x = 36sqr2x - 324

x+324 = 36sqr2x

square both sides

x^2 + 648x + 324^2 = 2x(36^2)

x^2 +648x + 104,976 = 2592x

x^2 +648x -2,592x + 104,976 = 0

x^2 - 1,944x + 104, 976 =0

use the quadratic formula

x = -b/2a +/- (1/2a)sqr(b^2 -4ac) with a = 1 b=-1944, c= 104976

x= about 5.4 or - 1939, but x can't be negative so reject 2nd "solution" as extraneous

check the answer 5.4 and it doesn't work mistake(s) somewhere above

sqr5.4 +14 - sqr10.8 + 5

about=19-1=18 not 1

try a graphing calculator

x=1684.4

check 1684.4

sqr1684.4+14-sqr3368.8+4

= 1

1684.4 works

but if you meant

sqr(x+14) - sqr(2x+5) = 1

then

x+14 = 1 - 2sqr(2x+5) + 2x+5

x -8 = 2sqr(2x+5)

x^2 -16x +64 = 8x+20

x^2 -24x =-44

x^2 -24x +144 = -44+144

(x-12)^2 = 10^2

x = 12 +/-10 = 2 or 22

but 22 is an extraneous solution not to the original problem

x = 2