🎯 Discriminant & Parameter Sweeps — Solution Counting

Essential technique for analyzing quadratic equations with parameters.

🎯 Recognition Cues

• “How many solutions” — Count real roots
• “For what values of $k$” — Parameter analysis
• “Find the range” — Parameter values that satisfy conditions
• Quadratic with parameter — $ax^2 + bx + c = 0$ where coefficients depend on parameter

📚 Template Solutions

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

🎯 Worked Examples

Example 1: Count Solutions

Problem: How many real solutions does $x^2 - 3x + 2 = 0$ have?

Solution:

1. Calculate discriminant: $\Delta = (-3)^2 - 4(1)(2) = 9 - 8 = 1$
2. Since $\Delta > 0$: There are 2 distinct real solutions
3. Answer: 2 solutions

Example 2: Parameter Range

Problem: 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 3: Parameter Range

Problem: 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: $m > 1$

⚠️ Common Pitfalls

Pitfall: 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

Pitfall: Incorrect inequality direction

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

Pitfall: 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 Patterns

• 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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