## anonymous one year ago What is the simplified form of 4x^2-25/2x-5?

1. anonymous

@Michele_Laino

2. Michele_Laino

hint: we can factorize the numerator as follows: $\Large \frac{{4{x^2} - 25}}{{2x - 5}} = \frac{{\left( {2x - 5} \right)\left( {2x + 5} \right)}}{{2x - 5}} = ...?$

3. anonymous

2x+5

4. Michele_Laino

since in general we have: $\Large {a^2}{x^2} - {b^2} = \left( {ax - b} \right)\left( {ax + b} \right)$

5. Michele_Laino

that's right!

6. anonymous

What would the restriction be?

7. Michele_Laino

since we can not divide by zero, then we have to exclude those values of x such that: $\Large 2x - 5 = 0$

8. anonymous

So -5/2

9. Michele_Laino

we have to exclòude x=5/2

10. Michele_Laino

exclude*

11. anonymous

So I was wrong when I said -5/2?

12. Michele_Laino

yes! since $2x - 5 = 0$ when: x=5/2

13. Michele_Laino

please try to solve that equation: 2x-5=0

14. anonymous

Ok! I see! So the answer would be 2x+5, with a restriction of 5/2

15. Michele_Laino

that's right!

16. anonymous

17. Michele_Laino

ok!

18. anonymous

What is the simplifed form of |dw:1437058472772:dw|

19. Michele_Laino

here we can write your division, as a multiplication of the first fraction, times the inverse of the second fraction, like below: $\Large \begin{gathered} \frac{{x + 1}}{{{x^2} + x - 6}}:\frac{{{x^2} + 5x + 4}}{{x - 2}} = \hfill \\ \hfill \\ = \frac{{x + 1}}{{{x^2} + x - 6}} \cdot \frac{{x - 2}}{{{x^2} + 5x + 4}} \hfill \\ \end{gathered}$

20. anonymous

yes!

21. Michele_Laino

now we have to factorize the subsequent polynomials: $\Large \begin{gathered} {x^2} + x - 6 \hfill \\ {x^2} + 5x + 4 \hfill \\ \end{gathered}$

22. anonymous

(x+2)(x-3) for the first one

23. Michele_Laino

ok! correct!

24. anonymous

(x+1)(x+4) for the second one

25. Michele_Laino

sorry, for first trinomial I got this: ${x^2} + x - 6 = \left( {x - 2} \right)\left( {x + 3} \right)$

26. anonymous

whoops yes:) you are right!

27. Michele_Laino

so, substituting our factorizations, in our computation, we get: $\Large \begin{gathered} \frac{{x + 1}}{{{x^2} + x - 6}}:\frac{{{x^2} + 5x + 4}}{{x - 2}} = \hfill \\ \hfill \\ = \frac{{x + 1}}{{{x^2} + x - 6}} \cdot \frac{{x - 2}}{{{x^2} + 5x + 4}} = \hfill \\ \hfill \\ = \frac{{x + 1}}{{\left( {x - 2} \right)\left( {x + 3} \right)}} \cdot \frac{{x - 2}}{{\left( {x + 1} \right)\left( {x + 4} \right)}} = ...? \hfill \\ \hfill \\ \end{gathered}$

28. anonymous

1/(x+3)(x+4)

29. Michele_Laino

that's right!

30. anonymous

Last one?

31. Michele_Laino

ok!

32. anonymous

Robin can mow a lawn in 3 hours, while Brady can mow the same lawn in 4 hours. How many hours would it take for them to mow the lawn together?

33. anonymous

a. 7/12 b. 12/7 c. 12 d. 7

34. Michele_Laino

If I call with W the work to be done, then the working rate of Robin is W/3, whereas the working rate of Brady is W/4

35. anonymous

Yes:)

36. Michele_Laino

now, when Robin and brady work together, then the working rate is: $\Large \frac{W}{3} + \frac{W}{4}$

37. Michele_Laino

38. Michele_Laino

so, we can write this equation: $\Large \left( {\frac{W}{3} + \frac{W}{4}} \right)t = W$ where t is the requested time

39. anonymous

I understand! Continue:) And then find like denominators

40. Michele_Laino

after a simplification at the left side, we can write: $\Large \frac{{4W + 3W}}{{12}}t = W$

41. Michele_Laino

42. anonymous

4x/12

43. Michele_Laino

hint: we can simplify the left side, so we get: $\Large \frac{{7W}}{{12}}t = W$

44. anonymous

:) Would that be the answer?

45. Michele_Laino

no, please we can simplify further that equation, and we get this: $\Large \frac{7}{{12}}t = 1$

46. Michele_Laino

now I multiply both sides by 12: |dw:1437059913299:dw|

47. anonymous

OH! Lol I'm sorry! I must have missed that.

48. anonymous

12/7 ?

49. Michele_Laino

that's right!

50. anonymous

Yay!!

51. Michele_Laino

:)

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