1. Start by moving -sqrt(x-5) to the right side of the equation.
sqrt(2x+1)-sqrt(x-5)=3
sqrt(2x+1)=3+sqrt(x-5)
2. Since both sides contain radical signs, square both sides.
(sqrt(2x+1))^2=(3+sqrt(x-5))^2
(sqrt(2x+1))(sqrt(2x+1))=(3+sqrt(x-5))(3+sqrt(x-5))
3. Simplify.
2x+1=color(red)9+6sqrt(x-5)color(blue)+(color(blue)x color(purple)(-5))
2x color(blue)(-x)+1 color(red)(-9) color(purple)(+5)=6sqrt(x-5)
x-3=6sqrt(x-5)
4. Since the radical sign still exists on the right side of the equation, square both sides again.
(x-3)^2=(6sqrt(x-5))^2
(x-3)(x-3)=(6sqrt(x-5))^2
5. Simplify.
x^2-6x+9=36(x-5)
x^2-6x+9=36x-180
6. Move all terms to the left side of the equation.
color(violet)1x^2 color(turquoise)(-42)x color(darkorange)(+189)=0
7. Use the quadratic formula to solve for x.
color(violet)(a=1)color(white)(XXX)color(turquoise)(b=-42)color(white)(XXX)color(darkorange)(c=189)
x=(-b+-sqrt(b^2-4ac))/(2a)
x=(-(color(turquoise)(-42))+-sqrt((color(turquoise)(-42))^2-4(color(violet)1)(color(darkorange)(189))))/(2(color(violet)1))
x=(42+-sqrt(1764-756))/2
x=(42+-sqrt(1008))/2
x=(42+-12sqrt(7))/2
x=(2(21+-6sqrt(7)))/(2(2))
x=(color(red)cancelcolor(black)2(21+-6sqrt(7)))/(color(red)cancelcolor(black)2(1))
color(green)(|bar(ul(color(white)(a/a)x=21+-6sqrt(7)color(white)(a/a)|)))
