## anonymous one year ago x^2 - 2x - 5/x - 3 ÷ x - 5/x^2 - 9

1. Nnesha

2. Nnesha

$\huge\rm \frac{ \frac{ x^2-2x-5 }{ x-3 } }{ \frac{ x-5 }{ x^2-9 }}$ and change division to multiplication to do that multiply top fraction with the RECIPROCAL of the bottom fraction example $\huge\rm \frac{ \frac{ a }{ b } }{ \frac{ c }{ d } }=\frac{ a }{ b } \times \frac{ d }{ c}$

3. Astrophysics

$\frac{ x^2-2x-5 }{ x-3 } \div \frac{ x-5 }{ x^2-9 }$ you need to factor the numerator $(x^2-2x-5)$ and remember when you divide by fractions you flip the second fraction and multiply, $\frac{ a }{ b } \div \frac{ c }{ d } \implies \frac{ a }{ b } \times \frac{ d }{ c }$

4. Nnesha

x^2-9 apply difference of squares rule $\huge\rm a^2-b^2 =(a+b)(a-b)$

5. Astrophysics

You will have to complete the square or use quadratic formula for x^2-2x-5

6. anonymous

I'm trying to keep up! Could you run me through the factoring process really quickly? I think I must be doing something wrong because my results don't match any of the possible answers.

7. Nnesha

are you sure its x^2-2x-5 ??

8. Astrophysics

Yeah I don't think that's right either haha

9. anonymous

I'm so sorry! It's x^2 - 2x - 15, not 5. But the rest should be right!

10. Nnesha

o^_^o

11. Astrophysics

Oh ok so now just find two numbers that add up to -2 and multiply together to give -15, can you think of two?

12. Nnesha

$$\color{blue}{\text{Originally Posted by}}$$ @Juliette2120 I'm trying to keep up! Could you run me through the factoring process really quickly? I think I must be doing something wrong because my results don't match any of the possible answers. $$\color{blue}{\text{End of Quote}}$$ show your work plz :)

13. anonymous

3 and - 5?

14. Astrophysics

Perfect!

15. myininaya

did you factor x^2-9 yet @Juliette2120

16. Astrophysics

So you have $\frac{ (x-5)(x+3) }{ x-3 } \times \frac{ x^2-9 }{ x-5 }$ now you can notice we can cancel out some terms, but lets first factor $x^2-9$ as nnesha mentioned here $$\color{blue}{\text{Originally Posted by}}$$ @Nnesha x^2-9 apply difference of squares rule $\huge\rm a^2-b^2 =(a+b)(a-b)$ $$\color{blue}{\text{End of Quote}}$$

17. myininaya

an example $x^2-25=(x-5)(x+5) \\ \text{ notice: I just replaced } a \text{ with } x \text{ and } b \text{ with 5 } \\ \text{ since } x^2-25=x^2-5^2$

18. myininaya

can you write 9 as a some number squared?

19. anonymous

@myininaya 3^2?

20. myininaya

right so a is x and b is 3 in this case

21. myininaya

$x^2-9=x^2-3^2 \\ x^2-3^2=(x-3)(x+3)$

22. anonymous

@Astrophysics I'm having a difficult time understanding the difference of squares rule! Could you explain that to me? Or show me how to factor this specific example?

23. anonymous

@myininaya Okay! I understand this. What's the next step?

24. Astrophysics

Well @myininaya just showed it but, it can be a bit tricky as it's not very intuitive to go from $a^2-b^2 \implies (a+b)(a-b)$ unless you distribute $(a+b)(a-b)$ itself

25. myininaya

$\frac{ x^2-2x-15 }{ x-3 } \div \frac{ x-5 }{ x^2-9 } \\ \frac{x^2-2x-15}{x-3} \times \frac{x^2-9}{x-5}$

26. Astrophysics

I guess for now just remember it, $a^2-b^2 \implies x^2-3^2$

27. myininaya

notice to change it to multiplication we just flip the second fraction

28. myininaya

now let's put in all of our factored forms

29. myininaya

$\frac{(x-5)(x+3)}{x-3} \cdot \frac{(x-3)(x+3)}{x-5} \\ \frac{(x-5)(x+3)(x-3)(x+3)}{(x-3)(x-5)}$ do you see anything that cancels?

30. anonymous

Do they have to be on the same level to cancel?

31. Nnesha

$$\color{blue}{\text{Originally Posted by}}$$ @Juliette2120 @Astrophysics I'm having a difficult time understanding the difference of squares rule! Could you explain that to me? Or show me how to factor this specific example? $$\color{blue}{\text{End of Quote}}$$ take square root of both terms (sqrt of 1st term + sqrt of 2nd term)(sqrt of 1st term - sqrt of 2nd term) easy to remember.

32. myininaya

you have to have a factor on top that matches a factor on bottom to cancel that common factor

33. anonymous

So it would be (x - 5) and (x - 3)?

34. Astrophysics

Nope you can always factor them out, also remember for example$\frac{ (x+1) }{ (x+1) } = 1$

35. Astrophysics

I guess I'll let @myininaya help you too much information all at once lol

36. anonymous

@Astrophysics I'm sorry about that! You've both been very helpful - it's all making a bit more sense!

37. myininaya

as you see you have an (x-5) on top and bottom so as @Astrophysics says (x-5)/(x-5)=1

38. myininaya

or in other words you can cancel the (x-5) on top with the one on bottom

39. myininaya

do you see anything else that can cancel?

40. anonymous

Is (x - 3) one?

41. myininaya

yes $\frac{(x-5)(x+3)}{x-3} \cdot \frac{(x-3)(x+3)}{x-5} \\ \frac{\cancel{(x-5)}(x+3)\cancel{(x-3)}(x+3)}{\cancel{(x-3)}\cancel{(x-5)}}$

42. myininaya

$\frac{(x+3)(x+3)}{1} \text{ or } (x+3)(x+3)$ do you know how to multiply (x+3)(x+3) out?

43. anonymous

So... It would be (x + 3)^2?

44. myininaya

yes that is right

45. myininaya

can we leave it as (x+3)^2?

46. myininaya

or do they want it in standard form?

47. anonymous

48. myininaya

k! do you want to talk more about the difference of squares formula?

49. anonymous

Trust me - I will be back shortly with more questions! I've been doing really well with math, but for some reason this module has me stumped!

50. myininaya

just in case you need something to look back on in the future: $(a-b)(a+b) =a(a+b)-b(a+b) \\ (a-b)(a+b)=a(a)+a(b)-b(a)-b(b) \\ (a-b)(a+b)=a^2+ab-ab-b^2 \\ (a-b)(a+b)=a^2-b^2 \\ \text{ Examples: } \\ x^2-1=(x-1)(x+1) \\ x^2-4=(x-2)(x+2) \\ x^2-9=(x-3)(x+3) \\ x^2-16=(x-4)(x+4) \\ x^2-25=(x-5)(x+5) \\ \text{ More Examples: } \\ 4x^2-25=(2x-5)(2x+5) \text{ note: I hope you see that } 4x^2=(2x)^2 \text{ and } 25=5^2$

51. anonymous

Thank you so much! I'll keep that!

52. anonymous

Thanks to all of you for your help! @myininaya @Astrophysics @Nnesha

53. myininaya

@Astrophysics and @Nnesha are the most awesomest! :) Yes I know that isn't a word. And @Juliette2120 you did great too.

54. Astrophysics

Haha, thanks and no problem! Everyone was great! XD

55. Nnesha

my pleasure.