5-Solving-Quadratic-Equations-and-Rational-Algebraic-Equations.pptx

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1. REVIEW Find a quadratic equation whose roots are .          

2. REVIEW Find a quadratic equation whose roots are .                  

3. Solving Quadratic Equations and Rational Algebraic Equations DOMINIC DALTON L. CALING Mathematics | Grade 9

4. Learning Objectives solves equations transformable to quadratic equations (including rational algebraic equations) solves problems involving quadratic equations and rational algebraic equations.

5. Solving Rational Algebraic Equations Transformable to Quadratic Equations STEPS: Multiply both sides of the equation by the LCD (Least Common Denominator). Write the resulting quadratic equation in standard form. Solve the resulting quadratic equation using any methods. (Extracting the Square Roots, Factoring, Completing the Square, Quadratic Formula)

6. EXAMPLE 1     Solution:        

7. SOLUTION   (solve using Factoring Method) ,        

8. EXAMPLE 2     Solution:        

9. SOLUTION   (solve using Factoring Method) ,      

10. EXAMPLE 3     Solution:          

11. SOLUTION   (solve using Factoring Method) ,      

12. Solving Problems Involving Rational Algebraic Equations Transformable to Quadratic Equations STEPS: Read, understand, and ANALYZE the problem. Identify the given and represent them in a variable or algebraic expressions. Formulate an equation Solve the equation. Check your answer. (OPTIONAL)

13. EXAMPLE 1 (WORK PROBLEM) Odette can finish washing a certain number of dishes in 15 minutes less than it takes Pam. If they work together, they can do the dishes in 10 minutes. How long will it take for them to finish the work separately?

14. SOLUTION Let 𝑤 = 1 whole work to be done. 𝑡 = time it takes for Pam to finish washing the dishes. time it takes for Odette to finish washing the dishes.   Rate (per minute)   Rate of Pam   Rate of Odette   Rate (together)  

15. SOLUTION Equation: (Rate of Pam) (Rate of Odette) (Rate together)       LCD:              

16. SOLUTION   (solve using Factoring Method)   ,   ,   Answer: Q: How long will it take for them to finish the work separately? time it takes for Odette to finish washing the dishes.   time it takes for Pam to finish washing the dishes.   It will take 30 minutes for Pam to finish the work. It will take 15 minutes for Odette to finish the work.

17. EXAMPLE 2 (WORK PROBLEM) A mason’s helper requires 4 hours more to pave a concrete walk than it takes the mason. The two worked together for 3 hours, then the mason was called away. The helper continued to work and completed the job in 2 hours. How long would it take each to do the same job if they work separately?

18. SOLUTION Let: x = number of hours it takes the mason to do the job x + 4 = number of hours it takes the helper to do the job part of the job done by the mason in one hour   part of the job done by the helper in one hour   Since the mason worked for 3 hours and the helper worked for 5 hours to complete the job, then the equation is  

19. Solution: LCD = x ( x + 4) 𝟑 + 𝟓 𝒙 = 𝟏 𝒙 𝒙 + 𝟒 𝒙+𝟒 𝟑 + 𝟓 𝒙 𝒙+𝟒 = 𝟏(𝒙)(𝒙 + 𝟒) 3( x + 4) + 5( x ) = x ( x + 4) 3 x + 12 + 5 x = x 2 + 4 x 8 x + 12 = x 2 + 4 x = x 2 + 4 x – 8 x – 12 = x 2 – 4 x – 12 = ( x – 6)( x + 2) Solve for x. x – 6 = x = 6 x + 2 = x = –2 We only get the positive value of x since the problem requires the number of hours. Therefore, the mason can do the job alone in 6 hours , while the helper can do the same job in 10 hours .

20. Example 3 : A rectangular table has an area of 27 m 2 and a perimeter of 24 m. What are the dimensions of the table? Solution : Let: l = length w = width P = 2 l + 2 w lw = A lw = 27 𝟐𝟒 = 𝟐 𝟐𝟕 𝒘 + 𝟐𝒘 → 𝒘 𝟐𝟒 = 𝟓𝟒 + 𝟐𝒘 𝒍 = 𝟐𝟕 𝒘 𝒘 𝒘 𝟐𝟒 = 𝟓𝟒 + 𝟐𝒘 𝒘 24 w = 54 + 2 w 2 = 2 w 2 – 24 w + 54 → w 2 – 12 w + 27 =

21. Example 4 : The sum of the squares of two consecutive integers is 85. What are integers? Solution : Let: x = first integer x + 1 = second integer Equation: x 2 + ( x + 1) 2 = 85 x 2 + x 2 + 2 x + 1 = 85 2 x 2 + 2 x + 1 – 85 = 2 x 2 + 2 x – 84 = x 2 + x – 42 = ( x + 7)( x – 6) = Solve for x. x + 7 = x = –7 x – 6 = x = 6 The two consecutive integers are 6 and 7 or –7 and –6. Check: (6) 2 + (7) 2 = 36 + 49 = 85 (-7) 2 + (-6) 2 = 49 + 36 = 85

22. Example 5 : A man drives 500 km to a business convention. On the return trip, he increases his speed by 25 km per hour and saves 1 hour driving time. How fast did he go on each trip? Solution : Let: x = speed in going to the convention x + 25 = speed of the return trip 𝟓𝟎𝟎 = length of time in going to the convention 𝒙 𝟓𝟎𝟎 𝒙+𝟐𝟓 = length of time in returning from the convention Equation: (length of time in going) = (length of time in returning) + 1 𝟓𝟎𝟎 𝟓𝟎𝟎 𝒙 𝒙 + 𝟐𝟓 = + 𝟏

23. Example 5 : Solution : LCD = x ( x + 25) 𝟓𝟎𝟎 = 𝟓𝟎𝟎 + 𝟏 𝒙 𝒙 + 𝟐𝟓 𝒙 𝟓𝟎𝟎 𝒙+𝟐𝟓 𝟓𝟎𝟎 𝒙 𝒙+𝟐𝟓 = + 𝟏 (𝒙)(𝒙 + 𝟐𝟓) 500( x + 25) = 500( x ) + 1( x )( x + 25) 500 x + 12, 500 = 500 x + x 2 + 25 x 500 x + 12, 500 = x 2 + 525 x = x 2 + 525 x – 500 x – 12, 500 0 = x 2 + 25 x – 12, 500 = ( x + 125)( x – 100) Solve for x. x + 125 = x = –125 x – 100 = x = 100 Speed is another measurement that cannot be negative. Therefore, the man drives 100 kph in going to the convention and returns at 125 kph.