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Discriminant and roots

1. Lesson 1.6 Discriminant and Roots of Quadratic Equations

2. At the end of the lesson, the learners should be able to do the following: Characterize the roots of a quadratic equation using the discriminant (M9AL-Ic-1) Describe the relationship between the coefficients and the roots of a quadratic equation (M9AL-Ic-2) .

3. Accurately solve the discriminant of a quadratic equation. Correctly determine the nature of the roots of a quadratic equation based on the value of its discriminant.

4. Correctly determine the value of missing coefficients so that quadratic equations will have one real root, two distinct roots, or imaginary roots.

5. Quadratic equations are used in a variety of real-life situations, such as determining the length and width of a rectangular field given its area, or the trajectory of an object.

6. You may have noticed that sometimes, the solutions that you obtain in a quadratic equation is not valid in the context of the word problem. For instance, you cannot have a negative value for measurements, or sometimes you obtain solutions containing imaginary numbers. It would be nice if there would be a test to determine if a solution contains imaginary numbers before you try answering them, right?

7. In fact, there is a way for you to check if the solution of a quadratic equation is imaginary or not without actually solving for it. In this scenario, you will solve for the discriminant of the quadratic equation. In this lesson, you will learn about the discriminant of a quadratic equation.

8. How does the discriminant determine the nature of the roots of quadratic equations? How will you find the values of missing coefficients so that quadratic equations have one real root, two distinct roots, or imaginary roots?

9. This is used to determine the nature of the roots of a quadratic equation. It is given by the formula   Discriminant

10. Example: The discriminant of is   Discriminant

11. Example: The discriminant of is   Discriminant

12. The discriminant can determine the nature of roots of a quadratic equation. It is determined as follows: If , the equation has two distinct real roots. If , the equation has one real root. If , the equation has no real roots. It has imaginary roots.   Nature of roots

13. Example: The discriminant of is . Since , the equation has one real root. The discriminant of is . Since , the equation has two distinct real roots.   Nature of roots

14. Example 1 : What is the discriminant of ?  

15. Example 1 : What is the discriminant of ?   Solution: Identify the values of , , and .  

16. Example 1 : What is the discriminant of ?   Solution: Substitute the obtained values of , , and into the formula for the discriminant.  

17. Example 1 : What is the discriminant of ?   Solution: Therefore, the discriminant of is .  

18. Example 2 : What is the nature of the roots of ?  

19. Example 2 : What is the nature of the roots of ?   Solution: Identify the values of , , and .  

20. Example 2 : What is the nature of the roots of ?   Solution: Substitute the obtained values of , , and into the formula for the discriminant.  

21. Example 2 : What is the nature of the roots of ?   Solution: Based on the value of , describe the roots of the given quadratic equation. Since , the roots of the quadratic equation are two distinct real numbers.  

22. Individual Practice: What is the value of the discriminant of the quadratic equation ? What is the nature of the roots of the quadratic equation ?  

23. Group Practice : To be done in two to five groups Describe the nature of the roots of the equation  

24. The d iscriminant ( ) is used to determine the nature of the roots of quadratic equations. It is given by the formula .  

25. The nature of roots can be determined by the discriminant as summarized below: If , the equation has two distinct real roots. If , the equation has one real root. If , the equation has no real roots. It has imaginary roots.  

26. Magsombol , Abigail T., et al. Global Mathematics . The Library Publishing House, Inc. 2015. Pierce, Rod. "Quadratic Equations" Math Is Fun. Ed. Rod Pierce. Retrieved 18 Jan 2019 from http://www.mathsisfun.com/algebra/quadratic-equation.html.