🎯 Equations with Parameters — Solution Counting & Analysis

Essential for AMC problems involving parameter analysis and solution counting.

🎯 Key Ideas

Parameter Analysis — Studying how the number and nature of solutions depends on parameter values.

Discriminant Method — Using $\Delta = b^2 - 4ac$ to analyze quadratic equations with parameters.

Degeneracy Conditions — Special cases where the equation becomes linear or has no solutions.

📊 Essential Techniques

Discriminant Analysis

Parameter Sweep Strategy

1. Identify the equation — Usually quadratic in form
2. Calculate discriminant — $\Delta = b^2 - 4ac$
3. Set up inequality — Based on desired number of solutions
4. Solve for parameter — Find range of parameter values
5. Check boundaries — Verify edge cases

🎯 Micro-Examples

Example 1: For what values of $k$ does $x^2 + kx + 1 = 0$ have exactly one real solution?

Solution:

1. Set up discriminant: $\Delta = k^2 - 4(1)(1) = k^2 - 4$
2. One solution when $\Delta = 0$: $k^2 - 4 = 0$
3. Solve: $k^2 = 4$, so $k = \pm 2$
4. Answer: $k = 2$ or $k = -2$

Example 2: Find all values of $m$ such that $mx^2 - 2x + 1 = 0$ has no real solutions.

Solution:

1. Set up discriminant: $\Delta = (-2)^2 - 4(m)(1) = 4 - 4m$
2. No solutions when $\Delta < 0$: $4 - 4m < 0$
3. Solve: $4m > 4$, so $m > 1$
4. Check $m = 0$: If $m = 0$, equation becomes $-2x + 1 = 0$, which has solution $x = \frac{1}{2}$
5. Answer: $m > 1$

Example 3: For what values of $a$ does the system $\begin{cases} x + y = a \ x^2 + y^2 = 1 \end{cases}$ have exactly one solution?

Solution:

1. From first equation: $y = a - x$
2. Substitute: $x^2 + (a-x)^2 = 1$
3. Expand: $x^2 + a^2 - 2ax + x^2 = 1$
4. Simplify: $2x^2 - 2ax + a^2 - 1 = 0$
5. Discriminant: $\Delta = (2a)^2 - 4(2)(a^2-1) = 4a^2 - 8a^2 + 8 = -4a^2 + 8$
6. One solution when $\Delta = 0$: $-4a^2 + 8 = 0$ → $a^2 = 2$ → $a = \pm\sqrt{2}$
7. Answer: $a = \sqrt{2}$ or $a = -\sqrt{2}$

⚠️ Common Traps & Fixes

Trap: Forgetting to check if leading coefficient is zero

• Fix: Always check if $a = 0$ in $ax^2 + bx + c = 0$
• Example: If $m = 0$ in $mx^2 - 2x + 1 = 0$, it’s not quadratic

Trap: Incorrect inequality direction

• Fix: Remember $\Delta > 0$ means 2 solutions, $\Delta < 0$ means 0 solutions
• Example: “No solutions” means $\Delta < 0$, not $\Delta > 0$

Trap: Missing edge cases

• Fix: Always check boundary values of parameters
• Example: When $m = 0$ in $mx^2 - 2x + 1 = 0$, it becomes linear

🎯 AMC-Style Worked Example

Problem: Find all real values of $k$ such that the equation $x^2 + (k-1)x + k = 0$ has two distinct real roots.

Solution:

1. Set up discriminant: $\Delta = (k-1)^2 - 4(1)(k) = k^2 - 2k + 1 - 4k = k^2 - 6k + 1$
2. Two distinct roots when $\Delta > 0$: $k^2 - 6k + 1 > 0$
3. Solve quadratic inequality: $k^2 - 6k + 1 = 0$ has roots $k = 3 \pm 2\sqrt{2}$
4. Since leading coefficient is positive: $k^2 - 6k + 1 > 0$ when $k < 3 - 2\sqrt{2}$ or $k > 3 + 2\sqrt{2}$
5. Answer: $k \in (-\infty, 3 - 2\sqrt{2}) \cup (3 + 2\sqrt{2}, \infty)$

Key insight: Parameter problems often require solving quadratic inequalities.

🔗 Related Topics

• Quadratic Equations — Discriminant is fundamental to quadratics
• Inequalities — Parameter analysis often involves inequalities
• Systems — Parameter analysis can involve systems
• Optimization — Finding parameter ranges for specific conditions

📝 Practice Checklist

• Master discriminant calculation
• Practice parameter range problems
• Learn to check edge cases
• Practice quadratic inequalities
• Understand solution counting
• Work on speed and accuracy

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