11. Start by moving -sqrt(x-5)−√x−5 to the right side of the equation.
sqrt(2x+1)-sqrt(x-5)=3√2x+1−√x−5=3
sqrt(2x+1)=3+sqrt(x-5)√2x+1=3+√x−5
22. Since both sides contain radical signs, square both sides.
(sqrt(2x+1))^2=(3+sqrt(x-5))^2(√2x+1)2=(3+√x−5)2
(sqrt(2x+1))(sqrt(2x+1))=(3+sqrt(x-5))(3+sqrt(x-5))(√2x+1)(√2x+1)=(3+√x−5)(3+√x−5)
33. Simplify.
2x+1=color(red)9+6sqrt(x-5)color(blue)+(color(blue)x2x+1=9+6√x−5+(x color(purple)(-5))−5)
2x2x color(blue)(-x)+1−x+1 color(red)(-9)−9 color(purple)(+5)=6sqrt(x-5)+5=6√x−5
x-3=6sqrt(x-5)x−3=6√x−5
44. 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)2=(6√x−5)2
(x-3)(x-3)=(6sqrt(x-5))^2(x−3)(x−3)=(6√x−5)2
55. Simplify.
x^2-6x+9=36(x-5)x2−6x+9=36(x−5)
x^2-6x+9=36x-180x2−6x+9=36x−180
66. Move all terms to the left side of the equation.
color(violet)1x^21x2 color(turquoise)(-42)x−42x color(darkorange)(+189)=0+189=0
77. Use the quadratic formula to solve for xx.
color(violet)(a=1)color(white)(XXX)color(turquoise)(b=-42)color(white)(XXX)color(darkorange)(c=189)a=1XXXb=−42XXXc=189
x=(-b+-sqrt(b^2-4ac))/(2a)x=−b±√b2−4ac2a
x=(-(color(turquoise)(-42))+-sqrt((color(turquoise)(-42))^2-4(color(violet)1)(color(darkorange)(189))))/(2(color(violet)1))x=−(−42)±√(−42)2−4(1)(189)2(1)
x=(42+-sqrt(1764-756))/2x=42±√1764−7562
x=(42+-sqrt(1008))/2x=42±√10082
x=(42+-12sqrt(7))/2x=42±12√72
x=(2(21+-6sqrt(7)))/(2(2))x=2(21±6√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)|)))
