🎯 Quadratics & Parabolas — Vertex, Discriminant & Completing Square
Quadratic equations and their graphical representations are central to AMC problems.
🎯 Key Ideas
Quadratic Formula — For $ax^2 + bx + c = 0$, the solutions are $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$. The discriminant $\Delta = b^2 - 4ac$ determines the nature of roots.
Vertex Form — The form $y = a(x-h)^2 + k$ reveals the vertex $(h,k)$ and axis of symmetry $x = h$. This is essential for optimization problems.
Completing the Square — Technique to convert $ax^2 + bx + c$ to vertex form, revealing maximum/minimum values and symmetry.
📊 Essential Formulas
| Concept | Formula | Usage |
|---|---|---|
| Quadratic formula | $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ | Solve any quadratic equation |
| Discriminant | $\Delta = b^2 - 4ac$ | Nature of roots |
| Vertex coordinates | $h = -\frac{b}{2a}$, $k = \frac{4ac-b^2}{4a}$ | From standard form |
| Vertex form | $y = a(x-h)^2 + k$ | Vertex at $(h,k)$ |
| Axis of symmetry | $x = -\frac{b}{2a}$ | Vertical line through vertex |
🔍 Discriminant Analysis
| Discriminant | Nature of Roots | Graph |
|---|---|---|
| $\Delta > 0$ | Two distinct real roots | Parabola crosses $x$-axis twice |
| $\Delta = 0$ | One repeated real root | Parabola touches $x$-axis once |
| $\Delta < 0$ | Two complex conjugate roots | Parabola doesn’t cross $x$-axis |
🎯 Micro-Examples
Example 1: Find vertex of $y = x^2 - 4x + 3$
- Method 1: Use formulas: $h = -\frac{-4}{2(1)} = 2$, $k = \frac{4(1)(3)-(-4)^2}{4(1)} = -1$
- Method 2: Complete the square: $y = (x-2)^2 - 1$, so vertex is $(2,-1)$
Example 2: Solve $x^2 - 5x + 6 = 0$
- Factoring: $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$
- Quadratic formula: $x = \frac{5 \pm \sqrt{25-24}}{2} = \frac{5 \pm 1}{2} = 2, 3$
Example 3: Find minimum value of $y = 2x^2 - 8x + 5$
- Complete the square: $y = 2(x^2 - 4x) + 5 = 2(x-2)^2 - 8 + 5 = 2(x-2)^2 - 3$
- Minimum value: $-3$ (occurs when $x = 2$)
⚠️ Common Traps & Fixes
Trap: Forgetting to factor out leading coefficient when completing the square
- Fix: Always factor out $a$ from $ax^2 + bx$ terms
- Example: $3x^2 - 6x + 1 = 3(x^2 - 2x) + 1 = 3(x-1)^2 - 3 + 1 = 3(x-1)^2 - 2$
Trap: Sign errors in vertex formula
- Fix: Remember $h = -\frac{b}{2a}$ (negative sign!)
- Example: For $x^2 + 4x + 3$, $h = -\frac{4}{2(1)} = -2$, not $2$
Trap: Confusing vertex and roots
- Fix: Vertex is $(h,k)$, roots are $x$-intercepts
- Example: $y = (x-2)^2 - 1$ has vertex $(2,-1)$ and roots $x = 1, 3$
🎯 AMC-Style Worked Example
Problem: For what value of $k$ does the equation $x^2 + 4x + k = 0$ have exactly one real solution?
Solution:
- Recognize: This is asking about the discriminant
- Set up: For exactly one solution, we need $\Delta = 0$
- Calculate: $\Delta = b^2 - 4ac = 4^2 - 4(1)(k) = 16 - 4k$
- Solve: $16 - 4k = 0$ gives $k = 4$
- Verify: When $k = 4$, we have $x^2 + 4x + 4 = (x+2)^2 = 0$, so $x = -2$ (one solution)
Key insight: The discriminant tells us about the nature of roots without solving.
🔗 Related Topics
- Polynomial Theory — Vieta’s formulas connect to quadratic roots
- Inequalities — Quadratic inequalities use similar techniques
- Systems — Nonlinear systems often involve quadratics
- Optimization — Vertex form is essential for max/min problems
📝 Practice Checklist
- Master quadratic formula and discriminant
- Practice completing the square
- Learn vertex form conversion
- Understand discriminant analysis
- Practice optimization problems
- Work on speed and accuracy
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