Answer:
x = 6 or x = 11
Step-by-step explanation:
Solve for x:
sqrt(x - 2) = 5 - sqrt(15 - x)
Raise both sides to the power of two:
x - 2 = (5 - sqrt(15 - x))^2
Subtract (5 - sqrt(15 - x))^2 from both sides:
-2 - (5 - sqrt(15 - x))^2 + x = 0
-2 - (5 - sqrt(15 - x))^2 + x = -42 + 10 sqrt(15 - x) + 2 x:
-42 + 10 sqrt(15 - x) + 2 x = 0
Simplify and substitute y = sqrt(15 - x).
-42 + 10 sqrt(15 - x) + 2 x = -12 + 10 sqrt(15 - x) - 2 (sqrt(15 - x))^2
= -2 y^2 + 10 y - 12:
-2 y^2 + 10 y - 12 = 0
The left hand side factors into a product with three terms:
-2 (y - 3) (y - 2) = 0
Divide both sides by -2:
(y - 3) (y - 2) = 0
Split into two equations:
y - 3 = 0 or y - 2 = 0
Add 3 to both sides:
y = 3 or y - 2 = 0
Substitute back for y = sqrt(15 - x):
sqrt(15 - x) = 3 or y - 2 = 0
Raise both sides to the power of two:
15 - x = 9 or y - 2 = 0
Subtract 15 from both sides:
-x = -6 or y - 2 = 0
Multiply both sides by -1:
x = 6 or y - 2 = 0
Add 2 to both sides:
x = 6 or y = 2
Substitute back for y = sqrt(15 - x):
x = 6 or sqrt(15 - x) = 2
Raise both sides to the power of two:
x = 6 or 15 - x = 4
Subtract 15 from both sides:
x = 6 or -x = -11
Multiply both sides by -1:
Answer: x = 6 or x = 11
