Hi! I’m your **AI Tutor and Learning Coach**. I’ll help you master **Quadratic Equations** step by step, using simple explanations, guided practice, and exam-focused questions. ## 🗺️ Learning Roadmap By the end, you should be able to: 1. Understand what a quadratic equation is. 2. Recognize the standard form: **ax² + bx + c = 0**. 3. Solve quadratics by: * Factorization * Completing the square * The quadratic formula 4. Understand the **discriminant** and what it tells us about solutions. 5. Solve word problems involving quadratic equations. 6. Recognize common exam question patterns and avoid common mistakes. ### Recommended sequence **Quadratic basics → Factorization → Solving by factorization → Completing the square → Quadratic formula → Discriminant → Word problems → Exam practice** --- # 📘 Concept 1: What Is a Quadratic Equation? A **quadratic equation** is an equation in which the highest power of the variable is **2**. For example: $$ x^2+5x+6=0 $$ This is quadratic because the highest power of \(x\) is 2. ### Standard Form A quadratic equation is usually written as: $$ \boxed{ax^2+bx+c=0} $$ where: * **\(a\)** = coefficient of \(x^2\) * **\(b\)** = coefficient of \(x\) * **\(c\)** = constant * **\(a \neq 0\)** ### Example Consider: $$ 2x^2+7x-4=0 $$ Compare it with: $$ ax^2+bx+c=0 $$ Therefore: $$ a=2,\qquad b=7,\qquad c=-4 $$ Notice that **\(c\) can be negative**. --- ## 🧠 Simple Analogy Think of the powers of \(x\) as levels: * \(x\) → Level 1 * \(x^2\) → Level 2 * \(x^3\) → Level 3 A **quadratic** is an equation whose highest level is **2**. So: $$ 3x^2+2x-5=0 $$ ✅ Quadratic But: $$ 3x^3+2x-5=0 $$ ❌ Not quadratic — it is a cubic equation. And: $$ 4x+7=0 $$ ❌ Not quadratic — it is linear. --- # ✏️ Check Your Understanding Don't look for answers yet—try them yourself. ### Question 1 Which of these is a quadratic equation? **A.** \(5x+2=0\) **B.** \(x^2-4x+3=0\) **C.** \(x^3+x-1=0\) ### Question 2 For: $$ 3x^2-8x+5=0 $$ identify: **a)** \(a\) **b)** \(b\) **c)** \(c\) ### Question 3 — Think Carefully Is this a quadratic equation? $$ 7-2x^2=0 $$ **Why or why not?** Reply with your answers to **1, 2, and 3**. I'll check them and then we'll move to **Concept 2: How to solve quadratic equations by factorization**.