Solving Quadratic Equations and Factorization

Various methods for solving quadratic equations, including factorization and the quadratic formula. Understand the concept of factorized forms and their significance in solving equations efficiently.

• Quadratic Equations
• Factorization
• Solving
• Mathematics

dano_win Follow

Uploaded on Feb 21, 2025 | 0 Views

Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.

Presentation Transcript

1. Function ? = Recap . Solutions when y = 0 ? = ? = and? = . Factorised form ? = .

2. What makes this example more difficult?

3. What if we zoom in? There isn t a factorised form of the function of this graph

4. How do we solve quadratic equations if we can t factorise? Show that it isn t possible to factorise this expression: ?2 3? + 4 Hint: Write down all the factors of 4

5. How do we solve quadratic equations if we can t factorise? So how can we solve the equation ?2 3? + 4 = 0 ? Today s lesson will show you how to use the quadratic formula to solve quadratic equations.

6. Here is a general quadratic expression: ??2+ ?? + ? The constant value is labelled ? The coefficient of ?2 is labelled ? The coefficient of ? is labelled ?

7. ?2+ 7 ? + 7? 4?2 1 > 0 Discuss: Which of these expressions are quadratic expressions? How do you know? 3?2+ 7?3+ ? ?2+ 7? 9 = 0 3?2+ 7? + 1

8. What is the value of ? in this expression? 3?2+ 2? + 4 3 2 4

9. What is the value of ? in this expression? 3?2+ 2? + 4 3 2 4

10. What is the value of ? in this expression? 3?2 2? + 4 2 -2 3

11. What is the value of ? in this expression? 4?2 2? + 6 6 -2 4

12. What is the value of ? in this expression? ?2 2? + 6 -1 0 1

13. What is the value of ? in this expression? 2? 3?2+ 6 -3 2 3

14. What is the value of ? in this expression? 2? 3?2 3 -3 2 3

15. What is the value of ? in this expression? 9 + 2? 3?2 -3 2 3

16. ?2+ 7 ? + 7? 4?2 1 > 0 Which of these expressions are quadratic? 3?2+ 7?3+ ? ?2+ 7? 9 = 0 3?2+ 7? + 1

17. Solving quadratic equations using the quadratic formula. This is the quadratic formula: ? = ? ?2 4?? 2? It enables us to find the solutions to the equation by using the values of ?, ? and ? from the quadratic equation.

18. Memory Game You will have 10 seconds to look at the quadratic equation You have to try and recreate as much of it as possible after this time has passed.

19. The quadratic formula. ? = ? ?2 4?? 2?

20. Solving quadratic equations using the quadratic formula. On whiteboards find the values of a, b and c for 2? + ? + 5 = 0 ? = 2 ? = 1 ? = 5

21. Solving quadratic equations using the quadratic formula. On whiteboards find the values of a, b and c for 7? 2? + 8 = 0 ? = 7 ? = -2 ? = 8

22. Solving quadratic equations using the quadratic formula. On whiteboards find the values of a, b and c for ? + 6? 3 = 0 ? = 1 ? = 6 ? = -3

23. Can you remember the quadratic formula? Write it on your boards. ? = ? ?2 4?? 2? We will now use this to solve the equations.

24. Using the quadratic formula Solve: ? + 9? + 4 = 0 ? = 1, ? = 9, ? = 4 Type the formula into your calculator replacing the letters with the values. ? = ? ?2 4?? 2? ? = ? ?2 4 ? ? 2 ?

25. How to type it into your calculator ? = ? ?2 4 ? ? 2 ? 1. Press the fraction button

26. How to type it into your calculator ? = ? ?2 4 ? ? 2 ? 1. Press the fraction button 2. Type in the numerator, but type in + , not .

27. How to type it into your calculator ? = ? ?2 4 ? ? 2 ? 1. Press the fraction button 2. Type in the numerator, but type in + , not . 3. To move to the denominator, click the down arrow. 4. Type in the denominator. 5. Press =

28. How to type it into your calculator ? = ? ?2 4 ? ? 2 ? Your calculator will give the answer 9 + 65 2 This is in surd form. To write as a decimal press the S D button. ? = 0.4688711259

29. How to type it into your calculator ? = ? ?2 4 ? ? 2 ? That is one of the solutions. To find the other, press the up arrow, move your cursor and change the + to a - Press = and S D You should get ? = 8.531128874

30. How to type it into your calculator ? = ? ?2 4 ? ? 2 ? The two solutions are ? = 0.4688711259 ? = 8.531128874 These would be the ?-intercepts for the graph of the function ? = ?2+ 9? + 4

31. Using the quadratic formula Solve: ? + 4? + 2 = 0 Now type into your calculator to find the two solutions. ? = 1, ? = 4, ? = 2 Type the formula into your calculator replacing the letters with the values. ? = ? ?2 4?? 2? ? = ? ?2 4 ? ? 2 ?

32. Using the quadratic formula Solve: 5? 7? 8 = 0 When the coefficient of ? is negative, extra care must be taken. Watch carefully as I go through the process on the board.

33. Using the quadratic formula Now let s look at some of your work.

34. For the questions on the next slide: Before answering each question, look at the previous answer. Do you think it will be related in any way? Check by working out. Were you correct? Predict. Check. Evaluate

37. Find and correct the common mistake