Question 1206468


{{{x + sqrt((x - 1/2) + sqrt(x + 1/4)) = 1024}}}


{{{sqrt((x - 1/2) + sqrt(x + 1/4)) = 1024-x}}}


{{{(sqrt((x - 1/2) + sqrt(x + 1/4)))^2 = (1024-x)^2}}}


{{{x - 1/2 + sqrt(x + 1/4) =x^2 - 2048 x + 1048576}}}


 {{{sqrt(x + 1/4) =x^2 - 2048x -x+ 1048576+1/2}}}


{{{(sqrt(x + 1/4))^2 =(x^2 - 2049 x + 2097153/2)^2}}}


{{{x + 1/4=x^4 - 4098 x^3 + 6295554 x^2 - 4297066497 x + 4398050705409/4}}}


{{{x^4 - 4098 x^3 + 6295554x^2 - 4297066497x + 4398050705409/4-x-1/4=0}}}


{{{x^4 - 4098 x^3 + 6295554x^2 - 4297066498x +1099512676352 =0}}}


using calculator we get

{{{x=992.02}}}
{{{x=993}}}= >will not work ({{{1025>1024}}})
{{{x≈1056.0 }}}= >will not work ({{{1056>1024}}})
{{{x=1057}}}= >will not work ({{{1057>1024}}})


 verifying solutions shows that only {{{x=992.02}}} works


{{{992.02 + sqrt((992.02 - 1/2) + sqrt(992.02 + 1/4))=1024}}} =>true

so, answer is

{{{x=992.02}}}