Solve for x:
x = 3 x sqrt(10-6 x)
Subtract 3 x sqrt(10-6 x) from both sides:
x-3 x sqrt(10-6 x) = 0
Simplify and substitute y = sqrt(10-6 x):
x-3 sqrt(10-6 x) x = 5/3-5 sqrt(10-6 x)-(sqrt(10-6 x)^2)/(6)+sqrt(10-6 x)^3/2 = y^3/2-y^2/6-5 y+5/3 = 0:
y^3/2-y^2/6-5 y+5/3 = 0
The left hand side factors into a product with three terms:
1/6 (3 y-1) (y^2-10) = 0
Multiply both sides by 6:
(3 y-1) (y^2-10) = 0
Split into two equations:
3 y-1 = 0 or y^2-10 = 0
Add 1 to both sides:
3 y = 1 or y^2-10 = 0
Divide both sides by 3:
y = 1/3 or y^2-10 = 0
Substitute back for y = sqrt(10-6 x):
sqrt(10-6 x) = 1/3 or y^2-10 = 0
Raise both sides to the power of two:
10-6 x = 1/9 or y^2-10 = 0
Subtract 10 from both sides:
-6 x = -89/9 or y^2-10 = 0
Divide both sides by -6:
x = 89/54 or y^2-10 = 0
Add 10 to both sides:
x = 89/54 or y^2 = 10
Take the square root of both sides:
x = 89/54 or y = sqrt(10) or y = -sqrt(10)
Substitute back for y = sqrt(10-6 x):
x = 89/54 or sqrt(10-6 x) = sqrt(10) or y = -sqrt(10)
Raise both sides to the power of two:
x = 89/54 or 10-6 x = 10 or y = -sqrt(10)
Subtract 10 from both sides:
x = 89/54 or -6 x = 0 or y = -sqrt(10)
Divide both sides by -6:
x = 89/54 or x = 0 or y = -sqrt(10)
Substitute back for y = sqrt(10-6 x):
x = 89/54 or x = 0 or sqrt(10-6 x) = -sqrt(10)
Raise both sides to the power of two:
x = 89/54 or x = 0 or 10-6 x = 10
Subtract 10 from both sides:
x = 89/54 or x = 0 or -6 x = 0
Divide both sides by -6:
Answer: |
| x = 89/54 or x = 0
