Solving Quadratic Equations Made Simple
Year 10 Mathematics Understanding quadratics step by step From basics to problem solving
What is a Quadratic Equation?
An equation where the highest power of x is 2 Standard form: ax² + bx + c = 0 Examples: x² + 5x + 6 = 0 or 2x² - 3x + 1 = 0 The graph is always a parabola (U-shape)
I DO: Solving by Factoring
Example: x² + 5x + 6 = 0 Step 1: Find two numbers that multiply to 6 and add to 5 Numbers: 2 and 3 (2 × 3 = 6, 2 + 3 = 5) Step 2: Write as (x + 2)(x + 3) = 0 Step 3: Solve x + 2 = 0 or x + 3 = 0 Solutions: x = -2 or x = -3
The Quadratic Formula
When factoring is difficult, use the formula x = (-b ± √(b² - 4ac)) / 2a Works for ANY quadratic equation The ± means there are usually two solutions
I DO: Using the Quadratic Formula
Example: 2x² - 7x + 3 = 0 Identify: a = 2, b = -7, c = 3 Substitute: x = (7 ± √(49 - 24)) / 4 Simplify: x = (7 ± √25) / 4 Calculate: x = (7 ± 5) / 4 Solutions: x = 3 or x = 1/2
WE DO: Practice Together
{"left":"Solve: x² - 4x - 5 = 0\nMethod 1: Factoring\nLook for factors of -5 that add to -4\nFactors: -5 and 1\n(x - 5)(x + 1) = 0\nSolutions: x = 5 or x = -1","right":"Method 2: Quadratic Formula\na = 1, b = -4, c = -5\nx = (4 ± √(16 + 20)) / 2\nx = (4 ± 6) / 2\nSolutions: x = 5 or x = -1"}
Quick Check Understanding
Which method would you use for x² + 6x + 9 = 0? A) Factoring (it's a perfect square) B) Quadratic formula C) Both work equally well Think about it and discuss with your partner!
YOU DO: Independent Practice
Solve these equations (choose your method): 1. x² + 7x + 12 = 0 2. 3x² - 5x - 2 = 0 3. x² - 6x + 9 = 0 Work in pairs and check each other's answers Solutions: 1) x = -3, -4 2) x = 2, -1/3 3) x = 3
Real-World Applications
Projectile motion (throwing a ball) Business profit calculations Area and geometry problems Engineering and physics Example: A ball's height h = -5t² + 20t + 1
Summary and Next Steps
Quadratic equations have the form ax² + bx + c = 0 Two main solving methods: factoring and formula Factoring is faster when possible Quadratic formula always works Practice makes perfect! Next lesson: Graphing quadratic functions