1. Quadratic Equations Chapter 4 Section 5, Section 7 and Section 8

2. 4-5 Quadratic Equations Page 2 ▪ Wherever the graph of a function f (x) intersects the x-axis, f (x) = 0. A value “x” for which f (x) = 0 is called a ZERO of the function. We also refer to this as the x- intercept. ▪ Essential Understanding – To find the zeros of a quadratic function y = ax 2 + bx + c, you MUST solve the related quadratic equation 0 = ax 2 + bx + c We do this by factoring the quadratic. We then apply the ZERO –Product Property. To Solve quadratic equations by factoring. To Solve quadratic equations by graphing.

3. ZERO-PRODUCT PROPERTY ▪ If a∙b = 0, then “a” has to be 0 OR “b” has to be 0. ▪ Think about it… ▪ Problem 1 Solving a Quadratic Equation by Factoring – What are the solutions of the quadratic equation

4. Got it? What are the solutions of the quadratic equation

5. Problem 2 Solving Quadratic Equations with Tables ▪ What are the solutions of the quadratic equation Remember to set the equation Equal to zero!! Then use diamond box method To factor

6. Problem 2 Solving Quadratic Equations with Tables ▪ What are the solutions of the quadratic equation

7. Problem 2 Solving Quadratic Equations with Tables ▪ What are the solutions of the quadratic equation

8. Problem 2 Solving Quadratic Equations with Tables Got it? ▪ What are the solutions of the quadratic equation Remember to set the equation Equal to zero!! Then use diamond box method To factor

9. Problem 2 Solving Quadratic Equations with Tables Got it? ▪ What are the solutions of the quadratic equation

10. Problem 3 – Solving a Quadratic by Graphing ▪ What are the solutions of the quadratic equation Remember to set the equation Equal to zero!! Then use diamond box method To factor

11. Problem 3 – Solving a Quadratic by Graphing ▪ What are the solutions of the quadratic equation

12. 4-5 Concept Byte Writing Equations from Roots To create quadratic equation based on the given roots. Standards A.CED.2

13. The solution of an equation is a value that makes the equation true. You can use the ZERO-Product Property to write a quadratic function from its zeros or quadratic equation from its roots. 1.Write a nonzero linear function f (x) that has a zero at x = 3 2.Write a nonzero linear function g (x) that has a zero at x = 4 3.For f and g from 1. and 2., write the product function h (x) = f (x) ∙ g (x) 4.What kind of function is h (x)?Solve the equation h (x) = 0

14. Mental Math!! Write a quadratic equation with each pair of values as roots.

15. You can also use zeros (or roots) to write quadratic expressions in standard form. 10.Complete the table shown. Write the product ( x – a ) ( x – b ) in standard form for each pair “a” and “b”. 11.Is there a pattern in the table? Explain. 12.If you know the roots, you can write a quadratic function or equation in standard form. Explain how this is possible. aba + ba ∙ b( x – a ) ( x – b ) 45920x 2 – 9x + 20 -451-20 4-5 -4-5 -9 -27 -9 -10 5 -20 20 9 -14 x 2 – 1x – 20 x 2 + 1x – 20 x 2 + 9x + 20 x 2 + 10x + 9 x 2 – 5x – 14

16. Exercises Find the sum and product of the roots for each quadratic equation 14. 2x 2 + 3x – 2 = 0 15. x 2 – 2x + 1 = 0 16. x 2 – 5x + 6 = 0

17. Exercises Find the sum and product of the roots for each quadratic equation 14. 2x 2 + 3x – 2 = 0 15. x 2 – 2x + 1 = 0 16. x 2 – 5x + 6 = 0

18. Exercises Given the sum and product of the roots, write the quadratic in standard form. 17. Sum is -3 Product is -18 18. Sum is 4 Product is 3 19. Sum is 2 Product is 0.75

19. 4-5 Algebra Review – Square Roots and Radicals

20. List the Perfect Squares from 1 to 20….. Things to remember: You CANNOT take the square root of a NEGATIVE number! Unless we are being asked about “imaginary numbers”…this will happen in the next section. Your solution to taking the square root of a number is always (±)

21. Examples

22. 4-7 The Quadratic Formula To solve equations using the Quadratic Formula. To determine the number of solutions by using the discriminant.

23. Essential Understanding – You can solve a quadratic equation ax 2 + bx + c in more than one way. In general, you can find a formula that give values of “x” in terms of a, b, and c. The Quadratic Formula is…

24. Problem 1 Using the Quadratic Formula What are the solutions?

26. Got it! Problem 1 Using the Quadratic Formula What are the solutions?

27. Got it?Problem 1 Using the Quadratic Formula What are the solutions?

28. Problem 2 Fundraising!

30. The Discriminant!!! ▪ A quadratic equation can have 2 real solutions ( x 2 = 4), one real solution ( x 2 = 0), or NO REAL solutions (x 2 = -4). ▪ In the Quadratic Formula, the part under the radical symbol b 2 -4ac, can tell us how many solutions exist. ▪ This is called the Discriminant.

32. Problem 3Using Discriminant What is the number of solutions of…

34. 4-8 Complex Numbers To identify and perform operations with complex numbers. To find complex numbers as solutions of quadratic equations.

35. In Chapter 1 we learned about different subset of real numbers. The set of real numbers is itself a subset of a larger set of numbers, the complex numbers.

36. Got it?How do you write the following using i?

37. A complex number is any number in the form a + bi where a and b are real numbers and b ≠ 0. Imaginary numbers and real numbers make up the set of complex numbers. Essential Understanding – You can define operations on the set of complex numbers so that when you restrict the operations to the subset of real numbers, you get the familiar operation on the real numbers. Problem 2Adding and Subtracting Complex Numbers

39. Problem 3 Multiplying Complex Numbers Remember i 2 = -1

41. Essential Understanding – Every quadratic equation has complex numbers as solutions (sometimes real sometimes not). Problem 5 Find the imaginary Solutions