Solve for x:
x^(1/4) + sqrt(x) = 12
Subtract 12 from both sides:
-12 + x^(1/4) + sqrt(x) = 0
Simplify and substitute y = x^(1/4).
-12 + x^(1/4) + sqrt(x) = -12 + x^(1/4) + (x^(1/4))^2
= y^2 + y - 12:
y^2 + y - 12 = 0
The left hand side factors into a product with two terms:
(y - 3) (y + 4) = 0
Split into two equations:
y - 3 = 0 or y + 4 = 0
Add 3 to both sides:
y = 3 or y + 4 = 0
Substitute back for y = x^(1/4):
x^(1/4) = 3 or y + 4 = 0
Raise both sides to the power of four:
x = 81 or y + 4 = 0
Subtract 4 from both sides:
x = 81 or y = -4
Substitute back for y = x^(1/4):
x = 81 or x^(1/4) = -4
Raise both sides to the power of four:
x = 81 or x = 256
x^(1/4) + sqrt(x) ⇒ 81^(1/4) + sqrt(81) = 12:
So this solution is correct
x^(1/4) + sqrt(x) ⇒ 256^(1/4) + sqrt(256) = 20:
So this solution is incorrect
The solution is:
x = 81
