2. 5.2 Solving Quadratic Equations by Factoring 5.3 Solving Quadratic Equations by Finding Square Roots

3. Objectives 1.Factor quadratic expressions and solve quadratic equations by factoring. 2.Find zeros of quadratic functions. Assignment: p 260 #24-30E, 36-40E, 66-74e Pg. 267 # 6-10, 12-17,60-68even, 71,72

4. Recall multiplying these binomials to get the standard form for the equation of a quadratic function: (x + 3)(x + 5) =+ 5x+3x+15 The “reverse” of this process is called factoring. Writing a trinomial as a product of two binomials is called factoring. (x + 3)(x + 5)

5. Factor Since the lead coefficient is 1, we need two numbers that multiply to –28 and add to –12. Factors of -28-1,281,-28-2,142,-14-4,74,-7 Sum of Factors 27-2712-123-3 Therefore:

6. Factor the expression: = (x-3)(x+7) Cannot be factored

7. Factoring a Trinomial when the lead coefficient is not 1. Factor: Multiply the coefficient of x 2 (a) by the constant term c to get a*c. Find two numbers whose sum is equal to the middle term and product equals a*c

9. Special Factoring Patterns you should remember: Pattern Name Pattern Example Difference of Two Squares Perfect Square Trinomial

10. Factor the quadratic expression:

11. A monomial is an expression that has only one term. As a first step to factoring, you should check to see whether the terms have a common monomial factor. Factor:

12. You can use factoring to solve certain quadratic equation. A quadratic equation in one variable can be written in the form where This is called the standard form of the equation: If this equation can be factored then we can use this zero product property. Zero Product Property Let A and B be real number or algebraic expressions. If AB = 0 the either A=0 or B=0

13. Solve: So, either (x+6)=0x = -6 Or (x – 3)=0x = 3 The solutions are –6 and 3. These solutions are also called zeros of the function Notice the zeros are the x-intercepts of the graph of the function.

14. Practice 1. x 2 -3x-4 =0 2. 5x 2 -13x+6=0 3. Find the zeros of the equation y = x 2 -4

15. Practice 1. x 2 -3x-4 =0 -1,4 2. 5x 2 -13x+6=0 3/5, 2 3. Write the equation in intercept form and find the zeros of the equation y = x 2 -4 (-2,2)

16. 5.3 Solving Quadratic Equations by Finding Square Roots (p. 264)

17. Falling Objects! Use h = -16t 2 + h 0 Height of the object after it has fallen # of seconds after the object is dropped Object’s initial height

18. Example The tallest building in the USA is in Chicago, Illinois. It is 1450 ft. tall. How long would it take a penny to drop from the top of the building to the ground?

19. How would you solve the equation: x 2 = 4 (take the square root of each side!) * Remember, the square root of a positive # has 2 answers! (one + and one -)

20. Radical Radical sign Radicand

21. Properties of Square Roots (a>0 and b>0) 1.Product Property – 2.Quotient Property- Example: Example:

22. Examples 1. 2. 3.

23. Rationalizing the Denominator You CANNOT leave a radical in the denominator of a fraction! No radicals in the basement!!!! (the numerator is OK) Just multiply the top & bottom of the fraction by the radical to “rationalize” the denominator.

24. More Examples! 1. 2. Can’t have a radical in the basement!

25. Solving Quadratic Equations 1.Solve. 3 - 5x 2 = -9 -3 -3 -5x 2 = -12 -5 -5 x 2 = 2.Solve. 3(x-2) 2 =21 3 3 (x-2) 2 = 7

26. More Examples! 3.Solve. 4x 2 -6=42 +6 +6 4x 2 =48 4 4 x 2 = 12 4. Solve.

27. Assignment