Solving Radical Equations (Lesson 5.4)

Introduction

• Objective: Get rid of the radicals in equations.
• Three-step process to solve radical equations.

Example 1: Solving Square Root Equations

1. Isolate the Radical
   • Move the radical expression to one side of the equation.
   • Example: ( \frac{\sqrt{x+1}}{2} = 2 )
   • Solution: Multiply both sides by 2 to cancel the fraction.
2. Square Both Sides
   • Square each side to eliminate the square root.
   • Example: ((\sqrt{x+1})^2 = 2^2 \rightarrow x+1 = 4)
3. Solve for x
   • Solve the resulting equation: (x = 3)
4. Check Your Solution
   • Substitute back to ensure the original equation holds true.

Example 2: Solving Cube Root Equations

1. Isolate the Cube Root
   • Move terms to isolate the cube root.
   • Example: (\sqrt[3]{2x-9} + 1 = 3)
   • Solution: Subtract 1 from both sides.
2. Cube Both Sides
   • Cube each side to eliminate the cube root.
   • Example: ((\sqrt[3]{2x-9})^3 = 3^3 \rightarrow 2x-9 = 27)
3. Solve for x
   • Solve the resulting equation: (x = 18)
4. Check Your Solution
   • Substitute back for verification.

Example 3: Equations with Radicals on Both Sides

• Isolate radicals on each side.
• Square both sides to eliminate the radicals.
• Solve the resulting linear equation.
• Check for extraneous solutions.

Example 4: Rational Exponents

1. Rational Exponents as Radicals
   • Treat rational exponents as radicals.
   • Example: (2x^{3/4} + 2 = 10)
   • Subtract 2 from both sides.
2. Raise to the Reciprocal Power
   • Eliminate the rational exponent by raising both sides to its reciprocal.
   • Example: ((2x)^{4/3} = 8^{4/3})
   • Use a calculator.
3. Solve for x
   • Simplify and solve: (x = 8)
4. Check Your Solution
   • Verify using substitution.

Example 5: Quadratic Equations Involving Radicals

1. Rational Exponents with Quadratics
   • Example: ((x+30)^{1/2} = x)
   • Raise to the power of 2 to eliminate the radical.
2. Rearrange to Form a Quadratic Equation
   • Example: (x^2 - x - 30 = 0)
3. Factor and Solve
   • Factor the quadratic to find solutions: (x = 6, x = -5)
4. Check for Extraneous Solutions
   • Verify by substituting back into the original equation.
   • Recognize (x = -5) as extraneous.

Conclusion

• Understanding the method of isolating radicals and checking for extraneous solutions is crucial.
• This lesson covers some of the toughest problems in solving radical equations.
• Begin homework assignments.