Deriving the quadratic formula
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Deriving the quadratic formula

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Deriving the quadratic formula from the quadratic equation

Deriving the quadratic formula from the quadratic equation

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Deriving the quadratic formula Presentation Transcript

  • 1. From Whence The Quadratic Formula? Don Simmons© D. T. Simmons, 2009 1
  • 2. Deriving the Quadratic FormulaEverybody taking algebra eventually learns the quadraticformula. But few know where it comes from.This presentation will help you understand how to get … From: ax 2 + bx + c = 0 −b ± b 2 − 4ac To: x= 2a© D. T. Simmons, 2009 2
  • 3. Deriving the Quadratic Formula• Begin with the quadratic equation… ax 2 + bx + c = 0• Set the leading coefficient to one. ax 2 + bx + c = 0 a b c x + x+ =0 2 a a© D. T. Simmons, 2009 3
  • 4. Deriving the Quadratic Formula• Isolate the variables on one side of the equation by moving the c value. b c x + x+ =0 2 a a b c x2 + x=− a a© D. T. Simmons, 2009 4
  • 5. Deriving the Quadratic Formula• Complete the square. b c x2 + x = − a a 2 2 b  b  c  b  x2 + x + ÷ = − + ÷ a  2a  a  2a © D. T. Simmons, 2009 5
  • 6. Deriving the Quadratic Formula• Create a common denominator. b b2 b 2 ( 4a ) c x2 + x + 2 = 2 − a 4a 4a ( 4a ) a• Combine the fractions b b2 b 2 − ( 4a ) c x2 + x + 2 = a 4a 4a 2© D. T. Simmons, 2009 6
  • 7. Deriving the Quadratic Formula• Rewrite in exponential form. b b2 b 2 − ( 4a ) c x2 + x + 2 = a 4a 4a 2 b 2 − ( 4a ) c 2  b   x + 2a ÷ =   4a 2© D. T. Simmons, 2009 7
  • 8. Deriving the Quadratic Formula• Apply the square root property b 2 − ( 4a ) c 2  b   x + 2a ÷ = ± 4a 2   b ± b 2 − ( 4a ) c x+ = 2a 2a© D. T. Simmons, 2009 8
  • 9. Deriving the Quadratic Formula• Isolate x. b ± b − ( 4a ) c 2 b b x+ − =− + 2a 2a 2a 2a −b ± b 2 − ( 4a ) c x= 2a© D. T. Simmons, 2009 9
  • 10. Deriving the Quadratic Formula• Done! −b ± b − ( 4a ) c 2 x= 2a© D. T. Simmons, 2009 10