Solving Radical Equations

Key Steps in Solving a Radical Equation

• Isolate the Radical: Make sure the radical is alone on one side of the equation.
• Raise Both Sides to the Power of the Index: This will remove the radical.
• Solve the Resulting Equation: After eliminating the radical, solve the equation as usual.
• Check Solutions: Substitute solutions back into the original equation to verify their validity.

Example 1

Equation: ( \sqrt{x + 6} = x + 4 )

1. Isolate the Radical: Radical is already isolated on the left.
2. Square Both Sides (index is 2):
   • Left side: ( (\sqrt{x + 6})^2 = x + 6 )
   • Right side: ( (x + 4)^2 = x^2 + 8x + 16 )
3. Solve Resulting Equation:
   • Equation becomes: ( x + 6 = x^2 + 8x + 16 )
   • Rearrange to form a quadratic: ( 0 = x^2 + 7x + 10 )
   • Factor: ( (x+5)(x+2) = 0 )
   • Solutions: ( x = -5 ) or ( x = -2 )
4. Check solutions:
   • For ( x = -5 ): ( \sqrt{-5+6} \neq -5+4 ) (Extraneous)
   • For ( x = -2 ): ( \sqrt{-2+6} = -2+4 ) (Valid)

Conclusion: Valid solution is ( x = -2 ).

Example 2

Equation: ( \sqrt{4x + 9} - 1 = x )

1. Isolate the Radical:
   • Add 1 to both sides: ( \sqrt{4x + 9} = x + 1 )
2. Square Both Sides:
   • Left side: ( 4x + 9 )
   • Right side: ( (x + 1)^2 = x^2 + 2x + 1 )
3. Solve Resulting Equation:
   • Equation becomes: ( 4x + 9 = x^2 + 2x + 1 )
   • Rearrange: ( 0 = x^2 - 2x - 8 )
   • Factor: ( (x-4)(x+2) = 0 )
   • Solutions: ( x = 4 ) or ( x = -2 )
4. Check solutions:
   • For ( x = 4 ): ( \sqrt{4\times4 + 9} - 1 = 4 ) (Valid)
   • For ( x = -2 ): ( \sqrt{4\times-2 + 9} - 1 \neq -2 ) (Extraneous)

Conclusion: Valid solution is ( x = 4 ).

Important Notes

• Extraneous Solutions: Solutions that do not satisfy the original equation once checked.
• Always check solutions in the context of the original equation to confirm validity.

This method is crucial for solving equations involving radicals and avoiding errors due to extraneous solutions.