Quadratic equations
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  • 1. Quadratic Equations
  • 2. A quadratic equation (in x) is an equation of the form 0 ,, 0 2 a cba cbxax with numbersrealareandwhere
  • 3. A quadratic equation has at most two (2) roots or solutions. These roots may be both real numbers or imaginary. If imaginary roots occur, they come in conjugate pairs.
  • 4. METHODS OF SOLVING QUADRATIC EQUATIONS by Square Root Property /Extracting the Root (for b = 0) by Factoring by Completing the Square by Quadratic Formula
  • 5. SOLVING QUADRATIC EQUATIONS BY SQUARE ROOT PROPERTY (for b = 0) 273.2 016.1 2 2 x x following.theSolve :Examples
  • 6. SOLVING QUADRATIC EQUATIONS BY FACTORING Some quadratic equations can be solved by factoring and using the following basic property of real numbers. 00 0 BA BA orifonlyandif PROPERTYPRODUCTZERO
  • 7. This means that if we can factor the left-hand side of a quadratic equation, then we can solve it by setting each factor equal to zero. This method works only when the right- hand side of the equation is zero.
  • 8. 344.5 456.4 0253.3 024.2 065.1 2 2 2 2 2 ww xx yy mm xx factoring.byequationsfollowingtheSolve :Examples
  • 9. SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE If a quadratic equation is of the form then we can solve it by taking the square root of each side. The left side will have a square root that is a linear expression in x. So, if a quadratic does not factor readily, then we can solve it by completing the square. cax 2 )(
  • 10. 22 2 2 2 22 . , 2 b x b bxx x b bxx square perfectthegivesThisoftcoefficien thehalfofsquaretheadd square,perfectamakeTo
  • 11. Examples: Solve the following by completing the square. 054.4 0182.3 02122.2 0138.1 2 2 2 2 xx xx yy xx
  • 12. SOLVING QUADRATIC EQUATIONS BY QUADRATIC FORMULA a acbb x a cbxax 2 4 0 0 2 2 are,where ,equation quadratictheofrootsThe
  • 13. Examples: Solve the following by quadratic formula. 037244.4 0968.3 0116.2 0473.1 2 2 2 2 xx xx xx xx