Slides on factoring Quadratic equation for middle school
- 2. UNIT 1 Overview UNIT TITLE Quadratic Equations KEY CONCEPT Form RELATED CONCEPT Patterns, Space, Equivalence GLOBAL CONTEXT Scientific and technical innovation STATEMENT OF INQUIRY Representing patterns with equivalent forms can lead to better systems, models and methods ATL SKILLS Social Skills (Collaboration skills), Research Skills (Information literacy skills)
- 4. A quadratic equation in one variable is an equation of degree 2. The standard form of a quadratic equation is , where a, b, and c are real numbers, and
- 5. Take note of this: •a is the coefficient of (cannot be zero) •b is the coefficient of x •c is the constant term
- 6. Some examples of quadratic equations are 4𝑥2 − 1 = 0 = 0 = 0 𝑥2 − 6𝑥 = 0
- 7. Quadratic Equations may be written in various forms. 50
- 8. Quadratic equations should always be written in standard form before solving, as this makes the process simpler and more organized.
- 9. Activity Write each quadratic equation in standard form and determine a, b, and c.
- 10. What is meant by finding the solution of the equation 4x + 6 =14? Let's Recall
- 11. •Any value that satisfies an equation is called a solution. •The solution is also called root. In solving quadratic equations, it means finding its solution(s) or root(s) that will satisfy the given equation.
- 12. WAYS of SOLVING QUADRATIC EQUATIONS EXTRACTING THE SQUARE ROOT FACTORISING COMPLETING THE SQUARE QUADRATIC FORMULA
- 13. Solving Quadratic Equations by Extracting the Square Roots
- 14. Learning Objectives determine the roots of a quadratic equation by extracting the square roots solve quadratic equation by extracting the square roots display accuracy in solving quadratic equations by extracting the square roots
- 16. A number that produces a specified quantity when multiplied by itself.
- 17. 3) Give the square roots of the following numbers. 2) – 4) 5)
- 18. we usually use this method to solve for x when the quadratic equations is in the form: Square Root Property
- 19. Square Root Property For any real number n If and then
- 20. Steps in Solving Quadratic Equations by Extracting the Square Roots 1. Transform the given in the form of 2. Simplify the equation so that only the term with variable remains on the left side of that equation 3. Extract the roots of both sides of the equation 4. Check your answer
- 21. 𝟏 . 𝒙𝟐 − 𝟏𝟔 =𝟎 Examples: 𝟐 . 𝒙 𝟐 = 𝟎 𝟑 . 𝒙𝟐 − 𝟗= 𝟎 𝟒 . ( 𝒙 − 𝟒)𝟐 − 𝟐𝟓=𝟎
- 22. STEP 1: Transform the given in the form of Example 1: 𝒙 𝟐 −𝟏𝟔=𝟎Apply Addition Property of Equality 𝒙𝟐 −𝟏𝟔+𝟏𝟔=𝟎+𝟏𝟔 𝒙 𝟐 =𝟏𝟔 Or simply transpose 16 to the other side 𝒙 𝟐 −𝟏𝟔=𝟎 𝒙 𝟐 =𝟏𝟔 Example 1: Example 1:
- 23. STEP 2: Simplify the equation so that only the term with variable remains on the left side of that equation Example 1: 𝒙 𝟐 =𝟏𝟔 Example 1:
- 24. STEP 3: Extract the roots of both sides of the equation Example 1: 𝒙 𝟐 =𝟏𝟔 √ 𝒙 𝟐 =± √𝟏𝟔 𝒙 =± 𝟒 ¿ 𝒙=+ 𝟒∧𝒙=−𝟒 Example 1:
- 25. STEP 4: Check your answer Example 1: 𝒙=𝟒 𝒙=− 𝟒 Checking: For 𝒙 𝟐 −𝟏𝟔=𝟎 (𝟒)𝟐 −𝟏𝟔=𝟎 𝟏𝟔−𝟏𝟔=𝟎 𝟎=𝟎 𝒙 𝟐 −𝟏𝟔=𝟎 For (− 𝟒)𝟐 −𝟏𝟔=𝟎 𝟏𝟔−𝟏𝟔=𝟎 𝟎=𝟎 Both values of x satisfy the given equation. So, the equation is true when x = 4 or when x= -4
- 26. Answer: The Equation has two solutions or roots: x= 4 or x= -4 Example 1:
- 27. Example 2: 𝒙 𝟐 = 𝟎 It is already in the form of Answer: Since Property No. 3
- 28. STEP 1: Transform the given in the form of Example 1: 𝒙 𝟐 − 𝟗=𝟎Apply Addition Property of Equality 𝒙 𝟐 −𝟗+ 𝟗=𝟎+𝟗 𝒙 𝟐 =𝟗 Or simply transpose 9 to the other side 𝒙 𝟐 =𝟗 Example 1: Example 3: 𝒙 𝟐 − 𝟗=𝟎
- 29. STEP 2: Simplify the equation so that only the term with variable remains on the left side of that equation 𝒙 𝟐 =𝟗 Example 1: Example 1: Example 3: Answer:
- 30. STEP 1: Transform the given in the form of Example 1: (𝒙 − 𝟒)𝟐 − 𝟐𝟓=𝟎 Apply Addition Property of Equality (𝒙 − 𝟒)𝟐 − 𝟐𝟓+𝟐𝟓=𝟎+𝟐𝟓 ( 𝒙 − 𝟒)𝟐 =𝟐𝟓 Or simply transpose -25 to the other side Example 1: Example 4: (𝒙 − 𝟒)𝟐 − 𝟐𝟓=𝟎 (𝒙 − 𝟒)𝟐 =𝟐𝟓
- 31. STEP 3: Extract the roots of both sides of the equation √(𝒙−𝟒) 𝟐 =±√𝟐𝟓 𝒙 − 𝟒=± 𝟓 ¿ 𝒙 − 𝟒=+𝟓∧𝒙 − 𝟒=− 𝟓 Example 1: Example 1: Example 4:
- 32. Answer: The Equation has two solutions or roots: x= 9 or x= -1 Example 4:
- 33. SOLVE Solve the following quadratic equation by extracting the square root. Show your solutions and box your final answer. (2 points each) 6. 7. 8. 9. 10.
- 35. Learning Objectives Identify the roots of a quadratic equations by factorising solve quadratic equation by factorising Show patience and accuracy in solving quadratic equations by factorising
- 36. Let's Recall Solve the following equations.
- 37. Let’s play a game
- 38. we usually use this method to solve for x when the quadratic equation is in the form: Factorisation
- 39. Steps in Solving Quadratic Equations by Factorising 1. Write the quadratic equation in standard form. Transform the given equation in the form: 2. Factor the quadratic expression on the left side of the equation. 3. Apply the Zero Product Property by equating the two factors of quadratic equation to 0. 5. Check your answers. 4. Find the values of the variable by solving the two equations
- 40. Zero Product Property The product AB = 0, if and only if A = 0 or B = 0.
- 41. How to Factor Quadratic Equations When a=1
- 42. 1. 𝑥 2 +6 𝑥 +8 =0 Examples: 3.
- 43. Factorize the given quadratic equation x² + 6x + 8 = 0 STEP 1: Identify the values of b and c. In this example, the values of b and c are: b=6 & c=8 STEP 2: Find two numbers that both ADD to b and both MULTIPLY to c. 2 and 4 works since the sum of 2 and 4 is 6 and the product of 2 and 4 is 8 STEP 3: Use your numbers from step two to write out the factors and solve.
- 44. Continuation Now we have a new equation: •(x+4)(x+2)=0 This is not our final answer. To solve this quadratic by factoring, we have to take each factor, set it equal to zero, and solve to find our solutions as follows: •x+4 = 0 x = -4 → •x+2 = 0 x = -2 →
- 45. Summary
- 46. Activity Solve the following quadratic equation by factorising. Show your solutions and box your final answer. 3. 4.
- 47. How to Factor Quadratic Equations When a≠1
- 48. 1. 3 𝑥2 +17 𝑥 +10=0 Examples: 3. 4.
- 49. Factorize the given quadratic equation STEP 1: Identify the values of a,b and c. In this example, the values of b and c are: a=3, b=17 & c=10 STEP 2: Multiply a and c first and then find two numbers that both ADD to b and both MULTIPLY to ac. ac = 3(10)=30. 2 and 15 works since the sum of 2 and 15 is 17 and the product of 2 and 15 is 30
- 50. continuation STEP 3: factor and replace the middle term with the two factors ↓
- 51. continuation STEP 4: Split the new quadratic down the middle and take the GCF of each side ↓ First Binomial: 3x² + 15x = 3x(x + 5) First Binomial: 2x + 10 = 2(x +5)
- 52. continuation STEP 4: Identify the Factors ↓ ↓ 3x(x + 5) ↓ (3x
- 53. Continuation Now we have a new equation: •(3x This is not our final answer. To solve this quadratic by factoring, we have to take each factor, set it equal to zero, and solve to find our solutions as follows: • 3x = 0 x = -2/3 → • = 0 x = -5 →
- 54. Find the solution of Find the solution of 2𝑥2 + 15𝑥 = −27 Each group must pick a problem to solve and present their answer to the class. Find the solution of + 7 Find the solution of + 3 Find the solution of Find the solution of+ 5
- 55. 1. What process are you going to use in solving quadratic equations if the given equations are in the form of ? 2. What process are you going to use in solving quadratic equations if the given equations are in the form of for a =1 and a ≠1? Generalizations: