🔢 Radical Equations — Isolate, Square, Verify

Essential technique for solving equations with square roots and other radicals.

🎯 Recognition Cues

• “Solve for $x$” — Equations with square roots
• “Find all real solutions” — Radical equations often have domain restrictions
• Square root expressions — $\sqrt{x}$, $\sqrt{x+1}$, etc.
• Multiple radicals — $\sqrt{x} + \sqrt{y} = a$

📚 Template Solutions

Basic Radical Equation

Multiple Radicals

🎯 Worked Examples

Example 1: Basic Radical Equation

Problem: Solve $\sqrt{x+1} = x-1$

Solution:

1. Isolate: Already isolated
2. Square: $x+1 = (x-1)^2 = x^2-2x+1$
3. Rearrange: $x^2-3x = 0$
4. Factor: $x(x-3) = 0$
5. Solve: $x = 0$ or $x = 3$
6. Check $x = 0$: $\sqrt{0+1} = \sqrt{1} = 1$ and $0-1 = -1$ ✗
7. Check $x = 3$: $\sqrt{3+1} = \sqrt{4} = 2$ and $3-1 = 2$ ✓
8. Answer: $x = 3$

Example 2: Multiple Radicals

Problem: Solve $\sqrt{x+1} + \sqrt{x-1} = 2$

Solution:

1. Domain: $x+1 \geq 0$ and $x-1 \geq 0$, so $x \geq 1$
2. Isolate one radical: $\sqrt{x+1} = 2 - \sqrt{x-1}$
3. Square both sides: $x+1 = 4 - 4\sqrt{x-1} + (x-1)$
4. Simplify: $x+1 = 4 - 4\sqrt{x-1} + x - 1 = 3 - 4\sqrt{x-1} + x$
5. Subtract $x$: $1 = 3 - 4\sqrt{x-1}$
6. Isolate radical: $4\sqrt{x-1} = 2$
7. Divide by 4: $\sqrt{x-1} = \frac{1}{2}$
8. Square: $x-1 = \frac{1}{4}$
9. Solve: $x = \frac{5}{4}$
10. Check domain: $\frac{5}{4} \geq 1$ ✓
11. Verify: $\sqrt{\frac{5}{4}+1} + \sqrt{\frac{5}{4}-1} = \sqrt{\frac{9}{4}} + \sqrt{\frac{1}{4}} = \frac{3}{2} + \frac{1}{2} = 2$ ✓
12. Answer: $x = \frac{5}{4}$

Example 3: Conjugate Technique

Problem: Solve $\sqrt{x+1} - \sqrt{x-1} = 1$

Solution:

1. Domain: $x+1 \geq 0$ and $x-1 \geq 0$, so $x \geq 1$
2. Multiply by conjugate: $(\sqrt{x+1} - \sqrt{x-1})(\sqrt{x+1} + \sqrt{x-1}) = 1 \cdot (\sqrt{x+1} + \sqrt{x-1})$
3. Simplify left side: $(x+1) - (x-1) = 2$
4. So: $2 = \sqrt{x+1} + \sqrt{x-1}$
5. Now we have: $\sqrt{x+1} - \sqrt{x-1} = 1$ and $\sqrt{x+1} + \sqrt{x-1} = 2$
6. Add equations: $2\sqrt{x+1} = 3$ → $\sqrt{x+1} = \frac{3}{2}$
7. Square: $x+1 = \frac{9}{4}$ → $x = \frac{5}{4}$
8. Check domain: $\frac{5}{4} \geq 1$ ✓
9. Verify: $\sqrt{\frac{5}{4}+1} - \sqrt{\frac{5}{4}-1} = \sqrt{\frac{9}{4}} - \sqrt{\frac{1}{4}} = \frac{3}{2} - \frac{1}{2} = 1$ ✓
10. Answer: $x = \frac{5}{4}$

⚠️ Common Pitfalls

Pitfall: Forgetting to check solutions

• Fix: Always substitute solutions back into original equation
• Example: $\sqrt{x+1} = x-1$ gives $x = 0, 3$, but only $x = 3$ works

Pitfall: Domain restrictions

• Fix: Check that radicands are non-negative
• Example: $\sqrt{x-1}$ requires $x \geq 1$

Pitfall: Incorrect squaring

• Fix: Square the entire expression, not just the radical
• Example: $(\sqrt{x+1} + \sqrt{x-1})^2 = (x+1) + 2\sqrt{(x+1)(x-1)} + (x-1)$, not $x+1 + x-1$

🎯 AMC-Style Worked Example

Problem: Find all real solutions to $\sqrt{x^2 + 1} = x + 1$.

Solution:

1. Domain: $x^2 + 1 \geq 0$ (always true for real $x$)
2. Square both sides: $x^2 + 1 = (x+1)^2 = x^2 + 2x + 1$
3. Simplify: $x^2 + 1 = x^2 + 2x + 1$
4. Subtract $x^2 + 1$: $0 = 2x$
5. Solve: $x = 0$
6. Check: $\sqrt{0^2 + 1} = \sqrt{1} = 1$ and $0 + 1 = 1$ ✓
7. Answer: $x = 0$

Key insight: Sometimes squaring eliminates the radical completely.

🔗 Related Patterns

• Domain Restrictions — Radical expressions have domain restrictions
• Extraneous Solutions — Squaring can introduce extra solutions
• Conjugates — Useful for certain radical expressions
• Verification — Always check solutions in original equation

📝 Practice Checklist

• Master basic radical equation solving
• Practice multiple radical techniques
• Learn conjugate method
• Practice domain restrictions
• Understand verification process
• Work on speed and accuracy

Next: Absolute Value Casework | Prev: Discriminant & Parameter Sweeps | Back: Problem Types Overview