To solve this equation, we need to isolate the variable x.
First, we will square both sides of the equation to get rid of the square roots:
Sqrt(x−2)+Sqrt(5)Sqrt(x-2) + Sqrt(5)Sqrt(x−2)+Sqrt(5)^2 = Sqrt202020^2x−2x-2x−2 + 2Sqrtx−2x-2x−2*Sqrt555 + 5 = 20
Now we can simplify the equation by expanding the left side:
x - 2 + 2Sqrt5(x−2)5(x-2)5(x−2) + 5 = 20x + 3 + 2Sqrt5x−105x - 105x−10 = 20
Next, we can isolate the square root term on one side of the equation and move the other terms to the other side:
2Sqrt5x−105x - 105x−10 = 20 - x - 32Sqrt5x−105x - 105x−10 = 17 - x
Now, square both sides to eliminate the square root term:
2Sqrt(5x−10)2Sqrt(5x - 10)2Sqrt(5x−10)^2 = 17−x17 - x17−x^245x−105x - 105x−10 = 17−x17 - x17−x17−x17 - x17−x 20x - 40 = 289 - 34x + x^2x^2 + 54x - 329 = 0
Now we have a quadratic equation that we can solve using the quadratic formula:
x = −54±sqrt(542−4(1)(−329))-54 ± sqrt(54^2 - 4(1)(-329))−54±sqrt(542−4(1)(−329)) / 2111 x = −54±sqrt(2916+1316)-54 ± sqrt(2916 + 1316)−54±sqrt(2916+1316) / 2x = −54±sqrt(4232)-54 ± sqrt(4232)−54±sqrt(4232) / 2x = −54±65.071-54 ± 65.071−54±65.071 / 2
Now we have two possible solutions for x:
x1 = −54+65.071-54 + 65.071−54+65.071 / 2 = 11.071x2 = −54−65.071-54 - 65.071−54−65.071 / 2 = -59.071
Therefore, the solutions to the equation are x = 11.071 and x = -59.071.
To solve this equation, we need to isolate the variable x.
First, we will square both sides of the equation to get rid of the square roots:
Sqrt(x−2)+Sqrt(5)Sqrt(x-2) + Sqrt(5)Sqrt(x−2)+Sqrt(5)^2 = Sqrt202020^2
x−2x-2x−2 + 2Sqrtx−2x-2x−2*Sqrt555 + 5 = 20
Now we can simplify the equation by expanding the left side:
x - 2 + 2Sqrt5(x−2)5(x-2)5(x−2) + 5 = 20
x + 3 + 2Sqrt5x−105x - 105x−10 = 20
Next, we can isolate the square root term on one side of the equation and move the other terms to the other side:
2Sqrt5x−105x - 105x−10 = 20 - x - 3
2Sqrt5x−105x - 105x−10 = 17 - x
Now, square both sides to eliminate the square root term:
2Sqrt(5x−10)2Sqrt(5x - 10)2Sqrt(5x−10)^2 = 17−x17 - x17−x^2
45x−105x - 105x−10 = 17−x17 - x17−x17−x17 - x17−x 20x - 40 = 289 - 34x + x^2
x^2 + 54x - 329 = 0
Now we have a quadratic equation that we can solve using the quadratic formula:
x = −54±sqrt(542−4(1)(−329))-54 ± sqrt(54^2 - 4(1)(-329))−54±sqrt(542−4(1)(−329)) / 2111 x = −54±sqrt(2916+1316)-54 ± sqrt(2916 + 1316)−54±sqrt(2916+1316) / 2
x = −54±sqrt(4232)-54 ± sqrt(4232)−54±sqrt(4232) / 2
x = −54±65.071-54 ± 65.071−54±65.071 / 2
Now we have two possible solutions for x:
x1 = −54+65.071-54 + 65.071−54+65.071 / 2 = 11.071
x2 = −54−65.071-54 - 65.071−54−65.071 / 2 = -59.071
Therefore, the solutions to the equation are x = 11.071 and x = -59.071.