Solving Radical Equations

Solve the Equation

• Given Equation:
  • The equation to solve is:
    $ext{√(√(√(2x + 6))) = (x - 9)}$

Steps to Solve the Equation

1. Raise Both Sides to the Power of 2:

   • To eliminate the outermost square root, square both sides:
     $ext{(√(√(√(2x + 6)))^2} = (x - 9)^2$
   • Simplifies to:
     $√(√(2x + 6)) = (x - 9)^2$

2. Square Again:

   • Square both sides again to eliminate the next square root:
     $ext{(√(2x + 6))^2 = ((x - 9)^2)^2}$
   • This leads to:
     $2x + 6 = (x - 9)^4$

3. Setting Up the Polynomial:

   • Rearranging gives us:
     $0 = (x - 9)^4 - 2x - 6$

4. Finding Roots:

   • This will require methods such as polynomial division or numerical methods to find roots of the polynomial equation.
   • If applicable, use a calculator to approximate the roots.

5. Verifying Solutions:

   • Once potential solutions are identified, substitute them back into the original equation to verify their correctness.

Final Result

• Solution:
  • The solution, once found, should be checked against the original equation to ensure it does not introduce extraneous roots.