SOLVING QUADRATIC EQUATIONS USING DIFFERENT WAYS.pptx
- 3. WAYS IN SOLVING QUADRATIC EQUATIONS EXTRACTING THE SQUARE ROOT FACTORING COMPLETING THE SQUARE QUADRATIC FORMULA
- 4. A solution to such an equation is called root/s Quadratic Equations can have two real solutions, one real solutions, or no real solution.
- 5. Solving Quadratic Equations by Extracting the Square Roots
- 6. 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
- 8. A number that produces a specified quantity when multiplied by itself.
- 9. 3) Give the square roots of the following numbers. 2) – 4) 5)
- 10. 1. How did you find each square root? 2. How many square root does a number have? Explain your answer. Questions: 3. Does a negative number have a square root? Why?
- 11. BING O Let’s play a game
- 12. If , when k is a real number, then Square Root Property
- 13. Roots/Solution of Quadratic Equation in Extracting the Square Roots 1. If , then has two real solutions or roots: 2. If , then has one real solution or root: x 3. If , then has no real solutions or roots.
- 14. 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
- 15. 𝟏 . 𝒙𝟐 − 𝟏𝟔 =𝟎 Examples: 𝟐 . 𝒎 𝟐 =𝟎 𝟑 . 𝒙𝟐 + 𝟗 =𝟎 𝟒 . ( 𝒙 − 𝟒)𝟐 − 𝟐𝟓=𝟎
- 16. 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:
- 17. 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 Since 16 is greater than 0, then has two real solutions or roots. (property 1) Example 1:
- 18. 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:
- 19. 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
- 20. Answer: The Equation has two solutions or roots: x= 4 or x= -4 Example 1:
- 21. Example 2: 𝒎𝟐 =𝟎 It is already in the form of Answer: Since Property No. 3
- 22. 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
- 23. STEP 2: Simplify the equation so that only the term with variable remains on the left side of that equation 𝒙 𝟐 =− 𝟗 Since -9 is less than 0, then (property No. 2) Example 1: Example 1: Example 3: Answer: The equation has no real solutions
- 24. 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
- 25. 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:
- 26. Find the solutions of the equation by extracting square roots. Find the solutions of the equation . Each group must pick a problem then allow them to solve and present their answer to the class. Solve the equation . Find the solutions of the equation .
- 27. 1. What process are you going to use in solving quadratic equations if the given equations are in the form of ? 2. How many solutions does a quadratic equation have in the form of if the k >0 ? k = 0? k <0 ? Generalizations:
- 28. 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.
- 29. 𝟏 . 𝒙+𝟕=𝟏𝟐 What would make a statement true? 𝟐 . 𝒕 − 𝟒=𝟎 𝟑 . 𝒓 +𝟓=−𝟑 𝟓 . 𝟐 𝒔=𝟏𝟔 𝟒 . 𝒙 −𝟏𝟎=− 𝟐 𝟔 . −𝟓 𝒙=𝟑𝟓 𝟕. 𝟑𝒉−𝟐=𝟏𝟔 𝟖 . −𝟕 𝒙=−𝟐𝟖 𝟏𝟎.𝟐(𝟑𝒌− 𝟏)=𝟐𝟖 𝟗 . 𝟑( 𝒙 +𝟕)=𝟐𝟒
- 31. Learning Objectives Identify the roots of a quadratic equations by factoring solve quadratic equation by factoring Show patience and accuracy in solving quadratic equations by factoring
- 32. Let's Recall Solve the following equations.
- 33. 4 𝑥 + 20 𝑥 2 Give the factors of the following polynomials. 𝑠2 + 8 𝑠 + 12
- 34. 1. How did you factor each polynomial? Answer the following questions: 3. How would you know if the factors you got are correct ones?
- 35. Steps in Solving Quadratic Equations by Factoring 1. Transform the given equation in the form of 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
- 36. If the product of two real number is zero, then either of two is equal to zero or both numbers are equal to zero (x)(y)=0 if and only if x= 0 or y = 0 Please take note that Zero Product Property will only be used if the right side of the equation is equal to zero. Zero Product Property
- 37. 1. 𝑥 2 +9 𝑥 =− 8 Examples: 2. 𝑡2 +12 𝑡 + 36= 0 3. 9 𝑥2 − 4 =− 8
- 38. Find the solution of Find the solution of Each group must pick a problem then allow them to solve and present their answer to the class. Find the solution of
- 39. Activity Solve the following quadratic equation by factoring. Show your solutions and box your final answer. (2 points each) 6. 7. 8. 9. 10.