Solving Quadratic Equations
A QUADRATIC equation has the form **ax² + bx + c = 0**. These equations model everything from projectile motion to profit optimization to the shape of a satellite dish. Mastering the three main solving techniques is essential for Algebra II, SAT math, and any STEM major.
Method 1: Factoring
When the quadratic factors nicely, use factoring.\n\nExample: x² - 5x + 6 = 0\n\nFactor: (x - 2)(x - 3) = 0\n\nZero Product Property: if a × b = 0, then a = 0 OR b = 0.\n\nSo x = 2, OR x = 3. Fast when it works.
Method 2: The Quadratic Formula
Works for EVERY quadratic. Memorize it:\n\n**x = (-b ± √(b² - 4ac)) / 2a**\n\nExample: 2x² + 3x - 2 = 0. Here a = 2, b = 3, c = -2.\n\nx = (-3 ± √(9 + 16)) / 4\nx = (-3 ± √25) / 4\nx = (-3 ± 5) / 4\n\nTwo solutions: x = 1/2, OR x = -2.
Method 3: Completing the Square
Rewrites the equation as a perfect square. Example: x² + 6x + 5 = 0\n\n1. Move constant: x² + 6x = -5\n2. Half of 6 is 3; square it: 9. Add to both sides: x² + 6x + 9 = 4\n3. Perfect square: (x + 3)² = 4\n4. Square root: x + 3 = ±2\n5. Solve: x = -1 OR x = -5\n\nEssential for vertex form and calculus prep.
What is b² - 4ac called?
When to Use Each Method
- Factoring — if coefficients are small integers\n- Quadratic formula — universal; use when factoring fails\n- Completing the square — for vertex form or deeper understanding
Graphing Quadratics
Quadratics graph as PARABOLAS. Key features:\n- Vertex (turning point)\n- Axis of symmetry: x = -b/(2a)\n- x-intercepts (where y = 0 — the solutions!)\n- y-intercept (where x = 0, equals c)\n\nEvery solution = where the parabola crosses the x-axis.
Solve x² - 4x - 5 = 0 by factoring.
Real-World Applications
- Projectile motion: y = -16t² + vt + h\n- Profit maximization\n- Area optimization\n- Bridge arches, satellite dishes\n- Falling objects, orbits
Solve 3 Ways
Solve x² - 2x - 3 = 0 by factoring, quadratic formula, and completing the square. All should give x = 3 or x = -1.
Real Problem
Ball thrown up at 40 ft/s from 5 ft: y = -16t² + 40t + 5. When does it hit ground? What is max height? Use quadratic formula and vertex formula.
Negative discriminant means how many real solutions?
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