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1. 1. Quadratic Equations
2. 2. A quadratic equation (in x) is an equation of the form 0 ,, 0 2 a cba cbxax with numbersrealareandwhere
3. 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. 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. 5. SOLVING QUADRATIC EQUATIONS BY SQUARE ROOT PROPERTY (for b = 0) 273.2 016.1 2 2 x x following.theSolve :Examples
6. 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. 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. 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. 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. 10. 22 2 2 2 22 . , 2 b x b bxx x b bxx square perfectthegivesThisoftcoefficien thehalfofsquaretheadd square,perfectamakeTo
11. 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. 12. SOLVING QUADRATIC EQUATIONS BY QUADRATIC FORMULA a acbb x a cbxax 2 4 0 0 2 2 are,where ,equation quadratictheofrootsThe
13. 13. Examples: Solve the following by quadratic formula. 037244.4 0968.3 0116.2 0473.1 2 2 2 2 xx xx xx xx