SOLUTION: Find x, x + √{(x - ½) + √(x + ¼)} = 1024

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Question 1206468: Find x,
x + √{(x - ½) + √(x + ¼)} = 1024
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

x+%2B+sqrt%28%28x+-+1%2F2%29+%2B+sqrt%28x+%2B+1%2F4%29%29+=+1024

sqrt%28%28x+-+1%2F2%29+%2B+sqrt%28x+%2B+1%2F4%29%29+=+1024-x

%28sqrt%28%28x+-+1%2F2%29+%2B+sqrt%28x+%2B+1%2F4%29%29%29%5E2+=+%281024-x%29%5E2

x+-+1%2F2+%2B+sqrt%28x+%2B+1%2F4%29+=x%5E2+-+2048+x+%2B+1048576

sqrt%28x+%2B+1%2F4%29+=x%5E2+-+2048x+-x%2B+1048576%2B1%2F2

%28sqrt%28x+%2B+1%2F4%29%29%5E2+=%28x%5E2+-+2049+x+%2B+2097153%2F2%29%5E2

x+%2B+1%2F4=x%5E4+-+4098+x%5E3+%2B+6295554+x%5E2+-+4297066497+x+%2B+4398050705409%2F4

x%5E4+-+4098+x%5E3+%2B+6295554x%5E2+-+4297066497x+%2B+4398050705409%2F4-x-1%2F4=0

x%5E4+-+4098+x%5E3+%2B+6295554x%5E2+-+4297066498x+%2B1099512676352+=0

using calculator we get
x=992.02
x=993= >will not work (1025%3E1024)
x%E2%89%881056.0+= >will not work (1056%3E1024)
x=1057= >will not work (1057%3E1024)

verifying solutions shows that only x=992.02 works

992.02+%2B+sqrt%28%28992.02+-+1%2F2%29+%2B+sqrt%28992.02+%2B+1%2F4%29%29=1024 =>true
so, answer is
x=992.02


Answer by ikleyn(53222) About Me  (Show Source):
You can put this solution on YOUR website!
.

Go to web-site

https://www.wolframalpha.com/widgets/gallery/view.jsp?id=b858339e64fa997454dd12f77cb1ece1


(wolfram alpha zeros calculator).


Print your expression for the function

    x + sqrt((x-0.5)+sqrt(x+0.25))-1024,

which represents  f(x) = x+%2B+sqrt%28%28x-0.5%29%2Bsqrt%28x%2B0.25%29%29-1024


and click the "Submit" button.  So, you want to get an approximation solution to this equation f(x) = 0.


You will get the number x = 992.02,  which is really a good approximation to the solution.


CHECK.  f(992.02) = 1024.005...

        f(992)    = 1023.984...


What I want to say at the end, is THIS: there is no Algebra method/methods
for finding/searching exact solution/solutions to this equation.

There are numerical methods, instead, that provide approximate solution.