Derivation of the Quadratic Formula

Introduction

• The quadratic formula is derived from the standard form of a quadratic equation.
• Standard form: ax² + bx + c = 0
• Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Steps to Derive the Quadratic Formula

1. Start with the standard form:

   • ax² + bx + c = 0

2. Transpose the constant:

   • ax² + bx = -c

3. Divide by the coefficient of x² (a):

   • (ax² + bx) / a = -c / a
   • Resulting in:
     • x² + (b/a)x = -c/a

4. Completing the Square:

   • Take the coefficient of x, which is (b/a),
   • Divide by 2: (b/2a),
   • Square it: (b/2a)² = b² / 4a²

5. Add this term to both sides:

   • x² + (b/a)x + (b²/4a²) = -c/a + (b²/4a²)

6. Left side becomes perfect square trinomial:

   • Factor: (x + b/2a)²

7. Simplify the right side:

   • Find common denominator (LCD = 4a²):
     • -c/a = -4ac/4a²
     • Combine:
       • (-4ac + b²) / 4a² = (b² - 4ac) / 4a²

8. Extract the square root:

   • x + b/2a = ± √((b² - 4ac)/4a²)
   • Simplifies to:
     • x + b/2a = ± (√(b² - 4ac) / 2a)

9. Isolate x:

   • x = -b/2a ± (√(b² - 4ac) / 2a)

10. Combine the terms:

    • Final formula:
      • x = (-b ± √(b² - 4ac)) / 2a

Conclusion

• Successfully derived the quadratic formula from the standard form of a quadratic equation.
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