2. Intro to Quadratics This is one of the hardest chapters in the book A quadratic equation is written in the form This is also known as Standard Form 13.1

3. Write in Standard Form and identify a, b, & c 13.1 A=4B=-7 C=-5 A=3 C=-7B=0

4. Solving ax 2 + bx = 0 Review factoring 6.1, 6.4, & 6.5 1.Factor by pulling out GCF 2.Set the GCF equal to zero and the parenthesis equal to zero. 3.Solve & Find 2 answers 13.1

5. Solving ax 2 + bx +c = 0 1.Put it in Standard Form 2.Factor by using the X-method or Guess & Check 3.Set both groups equal to zero. 4.Solve & Find 2 answers 13.1

6. You try… Solve: 13.1

7. Solving More Quadratics Solving If b=0 in a quadratic equation the easiest way to solve is to divide and take a square root 13.2a

8. Simplifying Radical Expressions Simplifying Square Roots 7 49 Radical Sign 13.2a

9. 1.Break down into Factor Tree 2.Rewrite & Pull out Pairs of Squares (bigger the better) 3.1 Innie Pair = Single Outie Simplifying Radicals 13.2a

10. You try… 13.2a

11. Fractions & Square Roots A radical can be split onto the top and bottom of a fraction 13.2a

12. Rationalizing the Denominator The final answer of a radical CANNOT contain a square root in the bottom of a fraction To get rid of it, multiply the top and bottom by the same radical that is in the denominator 13.2a

13. Don’t Forget to Simplify 13.2a

14. Steps to Solve ax 2 =k 1.Get the squared term by itself 2.The opposite of a variable squared is + and - square root 3.Simplify the square root & Rationalize the denominator 13.2a

15. You try.. 13.2a

16. Solving (x + a) 2 = k Steps to Solve 1.Try factoring into a binomial square 2.Take +/- square root of both sides 3.Solve for the variable 4.Simplify 13.2b

17. More examples 13.2b

18. You try… 13.2b

19. Projectiles An object that moves through the air and is only influenced by gravity Use this formula 13.2c Seconds Starting Height Initial Velocity Ending Height

20. Projectile Problem A marble is thrown upwards from height of 3m with an initial velocity of 14m/s. In about how many seconds will it hit the ground? 13.2c

21. Try it… A softball batter hits a pitch that is 2m above the ground. The ball pops up with an initial velocity of 9m/s. If the ball is allowed to hit the ground, how long will it be in the air? 13.2c

22. Interest Interest compounded annually Use this formula 13.2d Amount Earned Rate (percent) Principal (Initial Deposit) Time (years)

23. Example Find the interest rate if you made an initial deposit of $100, 2 years ago and now have $121 in your account. 13.2d

24. Completing the Square To complete the square we are going to Add Half of b 2 (middle coefficient squared) to both sides Then Factor 13.3

25. You try… Complete the square 13.3

26. Solving by Completing the square 1.Make a = 1 (Divide by “a”) 2.Get ax 2 +bx (space) = c 3.Add (b/2) 2 to both sides 4.Factor into a binomial square 5.Solve 13.3

27. More Examples… 13.3

28. You try… Solve by Completing the square 13.3

29. The Quadratic Formula Is derived from completing the square If ax 2 +bx+c=0 and a ≠ 0, then the quadratic formula is: 13.4

30. Proving the Quadratic Formula 13.4

31. Using The Quadratic Formula 1.Write out in Standard Form 2.Plug in values for a, b, & c. 3.Simplify (order of operations) 4.There cannot be roots of Neg. #’s Solve using the quadratic formula 13.4

32. You try… Solve by using the quadratic formula 13.4

33. The Discriminant The discriminant (what is inside the radical) can tell how many solutions there are If b 2 -4ac is +, there are 2 solutions If b 2 -4ac is 0, there is 1solution If b 2 -4ac is -, there are no solutions 13.4b

34. You try… Does the function intersect the x-axis in 0,1, or 2 places? And solve if possible. 13.4b

35. Solving Radical Equations The opposite of a radical (square root) is a square (exponent of 2) 13.6a

36. Steps to Solve 1.Get the radical (square root) by itself. Move everything else to the other side of the = sign 2.Square both sides 3.Solve using order of operations 13.6a

37. Try it… Etin-Osa! Solve 13.6a

38. Solving Radical Equations with 2 variables When there are variables squared on both sides you must check your solutions 13.6b

39. Steps to Solve 1.Get the radical (square root) by itself 2.Square both sides (FOIL if necessary) 3.Solve by putting into standard form and factoring 4.Check Answer (1 usually doesn’t work) 13.6b

40. Solving Formulas 1.Get the indicated variable by itself 2.Follow the same rules as numbers 3.The opposite of radicals are squares 13.6b

41. Pythagorean Theorem The side opposite the right angle is the “C-Side” (Hypotenuse = Longest Side) a b c 11.7 4= 3=

42. Steps to Solve 1.Plug in the hypotenuse (C side) 2.Then plug in the other 2 sides called legs 3.Simplify 4.The opposite of a square is a square root 5.Simplify the Square Root 11.7

43. X 7 7

44. 1 X

45. x 8

46. You try… 1.Find the hypotenuse if the legs are 2 and 6 2. Find a leg if the hypotenuse is 10 and a leg is 6. 11.7

47. Word Problems The hypotenuse of a right triangle is 25ft long. One leg is 17 feet longer than the other. Find the lengths of the legs 13.7 25 X+17 X

48. Word Problems A picture frame measures 20cm by 14cm. The picture inside the frame takes up 160 cm 2. Find the width of the Frame 13.7 20 14 w w 160 cm 2

49. Graph f(x) = -x 2 +2x+3 1)Use to find the x- coordinate of the vertex 2)The Axis of Symmetry is found by using 3)Choose 4 other values, 2 above and 2 below. Plug & Chug 12.4 Graphing y=ax 2 Basic Quadratic Functions with T-charts X 1243-4-3-2 Y 4 3 2 1 -3 -2 -4

50. Graph f(x) = -2x 2 First make a T-chart with 5 specific values 12.4 -8 -2 0 -8 Then Plot the points X 1243-4-3-2 Y 4 3 2 1 -3 -2 -4 Graphing y=ax 2 Basic Quadratic Functions with T-charts Exponents BEFORE subtraction

51. Graph f(x) = -x 2 12.4 Try It… -4 0 -4 X 1243 -3-2 Y 4 3 2 1 -3 -2 -4 Exponents before subtraction

52. Graph f(x) = -x 2 +2x+3 1)Use to find the x- coordinate of the vertex 2)The Axis of Symmetry is found by using 3)Choose 4 other values, 2 above and 2 below. Plug & Chug 12.4 Graphing y=ax 2 +bx+c Quadratic Functions with T-charts X 1243-4-3-2 Y 4 3 2 1 -3 -2 -4 4 3 0 3 0

53. Graph f(x) = -2x 2 +4x+1 1)Find the axis of symmetry 2)Find the coordinates of the vertex 3)Graph 12.4 Example X 1243-4-3-2 Y 4 3 2 1 -3 -2 -4 3 1 -5 1

54. Graph f(x) = x 2 -8x+16 1)Find the axis of symmetry 2)Find the coordinates of the vertex 3)Graph 12.4 You try… X 1243-4-3-2 Y 4 3 2 1 -3 -2 -4 0 1 4 1 4