Question 60914
The equation to solve is {{{sqrt(2x-1)+sqrt(x-4)=4}}}.

This is a quadratic equation in disguise!
The trick to solving these is to get all the square root stuff onto one side of the equation then square both sides. Then, you're left with a quadratic equation to solve:

Square both sides of the equation to solve:
{{{(sqrt(2x-1))^2 + sqrt(x-4)^2 = 4^2 = 16}}}

This expands to:
{{{(2x-1)+(x-4)+2sqrt((2x-1)(x-4)) = 16}}}

Collect like terms:
{{{3x-5+2sqrt((2x-1)(x-4)) = 16}}}

Subtract 16 from both sides:
{{{3x-21+2sqrt((2x-1)(x-4)) = 0}}}

Subtract {{{2sqrt((2x-1)(x-4))}}} from both sides:
{{{3x-21 = - 2sqrt((2x-1)(x-4))}}}

Now, square both sides:
{{{(3x-21)^2 = (-2sqrt((2x-1)(x-4)))^2}}}

Multiply things out:
{{{9x^2-126x+441 = 4(2x-1)(x-4)}}}

Or {{{9x^2-126x+441 = 4(2x^2-9x+4) = 8x^2-36x+16}}}

Subtract {{{8x^2}}} from both sides:
{{{x^2-126x+441=-36x+16}}}

Subtract 16 from both sides:
{{{x^2-126x+425=-36x}}}

Add 36x to both sides:
{{{x^2-90x+425=0}}}

This is a quadratic equation that you can solve with the quadratic formula or by factoring the equation if you can see the factors:

The equation factors into {{{(x-85)(x-5)}}}
The roots of the equation are 85 and 5

If you plug 85 into your original equation it doesn't work but 5 does so {{{5}}} is your answer.