Question 60914
{{{sqrt(2X-1)+sqrt(X-4)=4}}}   <----original equation

{{{sqrt(2X-1)=4-sqrt(X-4)}}}
{{{sqrt(2X-1)^2=(4-sqrt(X-4))^2}}}
{{{2X-1=(4-sqrt(X-4))(4-sqrt(X-4))}}}
{{{2x-1=4(4-sqrt(x-4))-sqrt(x-4)(4-sqrt(x-4))}}}
{{{2x-1=16-4sqrt(x-4)-4sqrt(x-4)+sqrt(x-4)^2}}}
{{{2x-1=16-8sqrt(x-4)+x-4}}}
{{{2x-1=-8sqrt(x-4)+x+12}}}
{{{2x-x-1-12=-8sqrt(x-4)+x-x+12-12}}}
{{{x-13=-8sqrt(x-4)}}}
{{{(x-13)^2=(-8sqrt(x-4))^2}}}
{{{x^2-26x+169=64(sqrt(x-4))^2}}}
{{{x^2-26x+169=64(x-4)}}}
{{{x^2-26x+169=64x-256}}}
{{{x^2-26x-64x+169+256=0}}}
{{{x^2-90x+425=0}}}
(x-5)(x-85)=0
x-5=0  and x-85=0
x=5  and x=85
Check for extraneous solutions:
Let x=5
{{{sqrt(2(5)-1)+sqrt((5)-4)=4}}}
{{{sqrt(10-1)+sqrt(5-4)=4}}}
{{{sqrt(9)+sqrt(1)=4}}}
{{{3+1=4}}}
4=4  x=5 is a valid solution.
let x=85
{{{sqrt(2(85)-1)+sqrt((85)-4)=4}}}
{{{sqrt(170-1)+sqrt(85-4)=4}}}
{{{sqrt(169)+sqrt(81)=4}}}
{{{13+9=4}}}
{{{22=4}}} x=85 is an extraneous solution, it's not valid.
Therefore the solution is: {{{highlight(x=5)}}}
Happy Calculating!!!