Sep 27, 2024

- Radical equations involve roots, such as square roots and cube roots.
- To solve, often isolate the radical on one side and then eliminate it by raising both sides to an appropriate power.
- Always check for extraneous solutions, as they can arise when squaring both sides.

**Equation:**( \sqrt{3x + 1} = 4 )**Steps:**- Square both sides: ( 3x + 1 = 16 )
- Subtract 1: ( 3x = 15 )
- Divide by 3: ( x = 5 )

**Verification:**Plug ( x = 5 ) back into original equation.

**Equation:**( \sqrt{7 - x} + 3 = 5 )**Steps:**- Subtract 3: ( \sqrt{7 - x} = 2 )
- Square both sides: ( 7 - x = 4 )
- Solve for ( x ): ( x = 3 )

**Verification:**Check by substitution.

**Equation:**( \sqrt[3]{x + 15} = 3 )**Steps:**- Cube both sides: ( x + 15 = 27 )
- Subtract 15: ( x = 12 )

**Equation:**( 2\sqrt{x} = x )**Steps:**- Square both sides: ( 4x = x^2 )
- Rearrange: ( x^2 - 4x = 0 )
- Factor: ( x(x - 4) = 0 )
- Solutions: ( x = 0 ) or ( x = 4 )

**Verification:**Check both solutions in the original equation.

**Equation:**( x^{1/4} + 4 = 7 )**Steps:**- Subtract 4: ( x^{1/4} = 3 )
- Raise to the 4th power: ( x = 81 )

**Equation:**( \sqrt{3x + 4} = \sqrt{4x + 3} )**Steps:**- Square both sides: ( 3x + 4 = 4x + 3 )
- Solve for ( x ): ( x = 1 )

- For equations with multiple radicals, isolate one radical before squaring.
**Equation:**( \sqrt{5 + x} + 2 = \sqrt{4x + 9} )**Steps:**- Move one radical: ( \sqrt{5 + x} = \sqrt{4x + 9} - 2 )
- Square both sides carefully to eliminate radicals.
- Simplify and solve the resulting polynomial equation.

**Isolate radicals:**Before squaring, isolate the radical expression.**Check solutions:**Always substitute back to verify, especially when dealing with potential extraneous solutions.**Factor polynomial equations:**Look for common factors or use the quadratic formula if factoring is complex.