1. The Quadratic Formula November 1, 2010

2. Quadratic Formula Methods to Solve Quadratics Equations Factoring But factoring is only “nice” when there are whole number answers and the real world doesn’t usually have “nice” answers. Graphing – not always exact answers Completing the Square Using the Quadratic Formula

5. Example 4-1a Use two methods to solve Method 1Factoring Original equation Factor Solve for x. Zero Product Property or

6. Example 4-1b Method 2Quadratic Formula For this equation, Quadratic Formula Multiply.

7. Example 4-1c Add. Simplify. or Answer: The solution set is {–5, 7}.

8. Example 4-1d Use two methods to solve Answer:

9. Example 4-2a Solve by using the Quadratic Formula. Round to the nearest tenth if necessary. Step 1Rewrite the equation in standard form. Original equation Subtract 4 from each side. Simplify.

10. Example 4-2b Step 2Apply the Quadratic Formula. Quadratic Formula and Multiply.

11. Example 4-2c Add. or

12. Example 4-2d Check the solutions by using the CALC menu on a graphing calculator to determine the zeros of the related quadratic function. Answer:The approximate solution set is {–0.3, 0.8}.

13. Example 4-2e Solveby using the Quadratic Formula. Round to the nearest tenth if necessary. Answer:

14. Discriminant The expression under the radical sign, b² – 4ac, is called the discriminant. The value of the discriminant can be used to determine the number of real roots for the quadratic equation.

16. State the value of the discriminant for. Then determine the number of real roots of the equation. and Simplify. Answer:The discriminant is –220. Since the discriminant is negative, the equation has no real roots.

17. Example 4-4b State the value of the discriminant for. Then determine the number of real roots of the equation. Step 1Rewrite the equation in standard form. Original equation Add 144 to each side. Simplify.

18. Example 4-4c Step 2Find the discriminant. Simplify. and Answer:The discriminant is 0. Since the discriminant is 0, the equation has one real root.

19. Example 4-4d State the value of the discriminant for. Then determine the number of real roots of the equation. Step 1Rewrite the equation in standard form. Original equation Subtract 12 from each side. Simplify.

20. Example 4-4e Step 2Find the discriminant. Answer:The discriminant is 244. Since the discriminant is positive, the equation has two real roots. Simplify. and

21. Example 4-4f State the value of the discriminant for each equation. Then determine the number of real roots for the equation. a. b. c. Answer:

22. Summary What are the different methods you know to solve quadratic equations? How do you use the Quadratic Formula? (“plug and chug”) What is the discriminant? How many real roots are there if the discriminant is positive? Negative? Zero? Assignment: