Quadratic Equations in One Variables.pptx

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1. Quadratic Equation in One Variable

2. Examples:   Quadratic Equation in One Variable A quadratic equation in one variable is of the form , where , , and are real number coefficients and . This equation is in the second degree wherein the highest exponent is 2.  

3.   Rewrite each equation in standard form. Then, identify the value of , , and .   1.   F O I L F: O: I: L:   Quadratic Equation:   T herefore, .  

4. Rewrite each equation in standard form. Then, identify the value of , , and .   2.   Quadratic Equation:     Therefore, .  

5. Rewrite each equation in standard form. Then, identify the value of , , and .   3.   Quadratic Equation:     Therefore, .  

6. Rewrite each equation in standard form. Then, identify the value of , , and .   DRILL  

7. The Roots of Quadratic Equations

8. Nature of the Roots If a, b, and c are real numbers and , then the roots of the quadratic equation are: real and unequal, if D 0; real and equal, if D 0; imaginary and unequal, if D 0; Moreover, the roots are: IV. rational and unequal, if D is a perfect square; V. irrational and unequal, if D is not a perfect square.   Discriminant The discriminant provided information regarding the nature of the roots of the quadratic equation.  

9. Determine the nature of the roots of each equation. 1.   ; a = 1, b = -6, and c = 9   Therefore, the roots of are real, equal, and rationa l .  

10. Determine the nature of the roots of each equation. 2.   ; a = 2, b = -4, and c = 5   Therefore, the roots of are imaginary and unequal.  

11. Determine the nature of the roots of each equation. DRILL  

12. Sum and Product of the Roots Sum of The Roots = ; Product of the Roots =   Find the sum and product of the roots of the following equations. 1.   ; a = 1, b = -6, and c = 9                

13. Find the sum and product of the roots of the following equations. 2.   ; a = 2, b = -4, and c = 5                

14. Determine the sum and products of the roots of each equation. DRILL  

15. Solving Quadratic Equations by: Extracting Square Roots Factoring Completing The Square Using The Quadratic Formula

16. No. of Solutions in a Quadratic E quation Every quadratic equation in one variable has exactly two solutions or roots.

17. Square Root Property If , then or .  

18. If , the equation is a pure or incomplete quadratic equation. Example: and If , the equation is a complete quadratic equation. Example: and   A. Extracting Square Roots Step 1: Express the quadratic equation in standard form. Step 2: Factor the quadratic expression. Step 3: Apply the zero-product property and set each variable factor equal to 0. Step 4: Solve the resulting linear equations.

19. Solve the quadratic equation by extracting the square root. 1.     To Check: If , If   True True  

20. Solve the quadratic equation by extracting the square root. 2.     To Check: If , If   True True  

21. Solve the quadratic equation by extracting the square root. 3.     To Check: If , If   True True   For , For ,  

22. Solve the following quadratic equations using the Square Root Property. DRILL  

23. Zero Property of Equality The product , if and only if or .  

24. B. Factoring Step 1: Clear all fractions (if any) and write the given equation in the form . Step 2: Factor the product of the coefficient of and the constant term of the given quadratic equation. Step 3: Express the coefficient of the middle term as the sum or difference of the factors obtained in Step 2. Step 4: Use zero product property to put each linear factor equal to 0. Step 5: Solve the resulting linear equations.  

25. Solve the quadratic equation by factoring. 1.             TO CHECK: True   TO CHECK: True  

26. Solve the quadratic equation by factoring. 2.             TO CHECK: True   TO CHECK: True  

27. Solve the following quadratic equations by factoring. DRILL  

28. C. Completing the Square Step 1: Rewrite the equation in the form . Step 2: Add to both sides the term needed to complete the square. Step 3: Factor the perfect square trinomial. Step 4: Solve the resulting equation by using the square root property..   To make an expression of the form a perfect square trinomial, add to it.  

29. Solve the quadratic equation by completing the square. 1.    

30. Solve the quadratic equation by completing the square. 1.   or   TO CHECK: True   TO CHECK: True   SOLUTIONS:  

31. Solve the quadratic equation by completing the square. 2.   or   SOLUTIONS:  

32. Solve the quadratic equation by completing the square. 3.    

33. Solve the quadratic equation by completing the square. 3.   Complete the Square: A dd the fractions on the right side.  

34. Solve the quadratic equation by completing the square. 3.    

35. Solve the quadratic equation by completing the square. 3.     SOLUTIONS:  

36. Solve the following quadratic equations by completing the square. DRILL  

37. D. Using the Quadratic Formula     Solve the quadratic equation using the quadratic formula. 1.  

38. Solve the quadratic equation using the quadratic formula. 1.    

39. Solve the quadratic equation using the quadratic formula. 1.   or or or  

40. Solve the following quadratic equations using the quadratic formula. DRILL  

41. Equations in Quadratic Form  

42. Equations in Quadratic Form An equation in quadratic form is any equation of the form , where is any algebraic expression. These equations can be solved using simple substitution.   Example: Solve the following equations. 1.   In , let .   or or  

43. Example: Solve the following equations. 1.   Let us solve for .   or or or or   Solution Set:  

44. Example: Solve the following equations. 2.   In , let .   or or  

45. Example: Solve the following equations. 2.   Let us solve for .   or or or   Solution Set:  

46. Solve for in each equation.   DRILL  

47. Rational Algebraic Equations Transformable to Quadratic Equation

48. Rational Algebraic Equations Rational algebraic equations are equations that contain one or more rational algebraic expressions (algebraic expressions in the form , where and are polynomials and ). These equations can be reduced to polynomial equations by clearing off fractions. Examples:  

49. But, take note, not all rational algebraic equations can be transformed into quadratic equations. Rational Algebraic Equations Transformable to Quadratic Equations It has to be a rational equation . There should only be one variable . If the expressions are linear, then there should be a variable on both numerator and denominator . Example: If the denominators are all constant, then the numerator should have at least one variable that has a power of 2 . Example:  

50. Solving Rational Algebraic Equations Transformable to Quadratic Equations by Finding the LCD Multiply both sides of the equation by the LCD . Apply the distributive law . Combine like terms . Apply the inverse operation . Rewrite the equation in standard form. Solve for the resulting quadratic equation using any of the 4 methods.

51. Example: Solve the following equations. 1.       [ ]   [ ]    

52. Example: Solve the following equations. 1.    or or  

53. Solving Rational Algebraic Equations Transformable to Quadratic Equations using Cross-Multiplication Cross-multiply the means and the extremes. Apply the inverse operation . Rewrite the equation in standard form. Solve for the resulting quadratic equation using any of the 4 methods. Cross-Multiplication Property: If , then .   then .  

54. Example: Solve the following equations. 1.         EXTREMES MEANS EXTREMES MEANS

55. Example: Solve the following equations. 1.    or or or  

56. Solve the following equations. DRILL  

57. Problem Solving involving Quadratic Equations

58. PROBLEM # 1: The sum of two numbers is 22 and the sum of their squares is 250. Find the numbers. STEP 1: Define the variables. Let one of the two numbers the other number   STEP 2: Identify the thought process. one of the two numbers 2 + the other number 2 = 250

59. PROBLEM # 1: The sum of two numbers is 22 and the sum of their squares is 250. Find the numbers. STEP 3: Formulate the equation.   STEP 4: Solve the equation.    

60. PROBLEM # 1: The sum of two numbers is 22 and the sum of their squares is 250. Find the numbers. STEP 4: Solve the equation.     The numbers are 9 and 13. To Check:  

61. Solve the word problem below. DRILL PROBLEM: Two numbers differ by 9. The sum of their squares is 653. Identify the 2 numbers. Let the smaller number the larger number  

62. PROBLEM # 2: If the length of each side of a square is increased by 5 cm, the area is multiplied by 4. What is the length of the original side of the square? STEP 1: Define the variables. Let one of the two numbers the other number   STEP 2: Identify the thought process. one of the two numbers + 5 2 = 4 area of the square

63. PROBLEM # 2: If the length of each side of a square is increased by 5 cm , the area is multiplied by 4. What is the length of the original side of the square? STEP 3: Formulate the equation.   STEP 4: Solve the equation.  

64. PROBLEM # 2: If the length of each side of a square is increased by 5 cm , the area is multiplied by 4. What is the length of the original side of the square? STEP 4: Solve the equation.   Since a side of the square is a physical quantity that cannot assume a negative value, we can only accept . Therefore, the length of the original side of the square is 5 cm.  

65. PROBLEM # 3: A rectangle is 30 cm long and 20 cm wide. A rectangular strip added to one side and another of the same width to the other side result in the doubling of the area. Find the width of the strip. STEP 1: Draw the figure.

66. PROBLEM # 3: A rectangle is 30 cm long and 20 cm wide. A rectangular strip added to one side and another of the same width to the other side result in the doubling of the area. Find the width of the strip. STEP 3: Formulate the equation.   STEP 4: Solve the equation.   Let width of the strip   STEP 2: Define the variable.

67. STEP 4: Solve the equation.   Since the width cannot be negative, we can only accept . Therefore, the width of the rectangular strip is 10 cm.   PROBLEM # 3: A rectangle is 30 cm long and 20 cm wide. A rectangular strip added to one side and another of the same width to the other side result in the doubling of the area. Find the width of the strip.

68. Solve the word problem below. DRILL PROBLEM: The perimeter of a rectangle is 60 m and its area is . Find the length and the width of the rectangle.