Solve for x:
2 + sqrt(2 x + 3) = sqrt(6 x + 7)
Raise both sides to the power of 2:
(2 + sqrt(2 x + 3))^2 = 6 x + 7
Subtract 6 x + 7 from both sides:
-7 - 6 x + (2 + sqrt(2 x + 3))^2 = 0
-7 - 6 x + (2 + sqrt(2 x + 3))^2 = 4 sqrt(2 x + 3) - 4 x:
4 sqrt(2 x + 3) - 4 x = 0
Simplify and substitute y = sqrt(2 x + 3):
4 sqrt(2 x + 3) - 4 x = 6 + 4 sqrt(2 x + 3) - 2 (sqrt(2 x + 3))^2 = -2 y^2 + 4 y + 6 = 0:
-2 y^2 + 4 y + 6 = 0
The left hand side factors into a product with three terms:
-2 (y - 3) (y + 1) = 0
Divide both sides by -2:
(y - 3) (y + 1) = 0
Split into two equations:
y - 3 = 0 or y + 1 = 0
Add 3 to both sides:
y = 3 or y + 1 = 0
Substitute back for y = sqrt(2 x + 3):
sqrt(2 x + 3) = 3 or y + 1 = 0
Raise both sides to the power of two:
2 x + 3 = 9 or y + 1 = 0
Subtract 3 from both sides:
2 x = 6 or y + 1 = 0
Divide both sides by 2:
x = 3 or y + 1 = 0
Subtract 1 from both sides:
x = 3 or y = -1
Substitute back for y = sqrt(2 x + 3):
x = 3 or sqrt(2 x + 3) = -1
Raise both sides to the power of two:
x = 3 or 2 x + 3 = 1
Subtract 3 from both sides:
x = 3 or 2 x = -2
Divide both sides by 2:
x = 3 or x = -1
2 + sqrt(2 x + 3) ⇒ 2 + sqrt(3 + 2 (-1)) = 3
sqrt(6 x + 7) ⇒ sqrt(7 + 6 (-1)) = 1:
So this solution is incorrect
2 + sqrt(2 x + 3) ⇒ 2 + sqrt(3 + 2 3) = 5
sqrt(6 x + 7) ⇒ sqrt(7 + 6 3) = 5:
So this solution is correct
The solution is:
Answer: |x = 3
