## sloppycanada one year ago How would one go about solving the following rational functions?

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2. anonymous

Equals what again ?

-2

@AngusV

5. SolomonZelman

Can you draw your function a little better, please? I can see the left hand side of the equation, $$\large\color{black}{ \displaystyle \frac{5}{5-p}-\frac{p^2}{5-p} }$$ but what is this equal to? What is the right side of your equation?

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7. SolomonZelman

$$\large\color{black}{ \displaystyle \frac{5}{5-p}-\frac{p^2}{5-p}=-2}$$ combine the two fractions first

8. SolomonZelman

$$\large\color{black}{ \displaystyle \frac{5-p^2}{5-p}=-2}$$ would this be correct, how do you think?

No this wouldn't be correct? Since -p - (-p) would cancel out right?

10. SolomonZelman

it is correct

11. SolomonZelman

just as $$\large\color{black}{ \displaystyle \frac{A}{B}-\frac{C}{B}\Rightarrow \frac{A-C}{B}}$$ so is $$\large\color{black}{ \displaystyle \frac{5}{5-p}-\frac{p^2}{5-p}\Rightarrow \frac{5-p^2}{5-p}}$$

12. SolomonZelman

is this making sense?

Oh okay.. so then I multiply both sides by the denominator?

14. SolomonZelman

yes, you need to multiply times 5-p on both sides

15. SolomonZelman

$$\large\color{black}{ \displaystyle \frac{5-p^2}{5-p}=-2}$$ $$\large\color{black}{ \displaystyle \left(\frac{5-p^2}{5-p}\right)\times (5-p)=-2 \times (5-p)}$$

16. SolomonZelman

5-p on the left side cancels $$\large\color{black}{ \displaystyle 5-p^2=-2(5-p)}$$

17. SolomonZelman

now, solve for p

5 - p^2 = -10 + 2p?

19. SolomonZelman

yes

20. SolomonZelman

now, bring everything over to the left side

21. SolomonZelman

or, you can subtract 2p from both sides, subtract 5 from both sides, divide by -1 on both sides, and complete the square

5-p^2 + 10 - 2p = 0 15p-p^2 -2p

23. SolomonZelman

15p?

Sorry 15 - p^2

25. SolomonZelman

where is -2p though?

26. SolomonZelman

should be: -p²-2p+15=0

15 - p^2 - 2p = 0?

oh okay

29. SolomonZelman

yes, in my order is going to be more convenient

30. SolomonZelman

-p²-2p+15=0 divide both sides by -1, and factor the left side

wait -5 or 3

32. SolomonZelman

you mean the numbers to factor, or solutions for p?

33. SolomonZelman

p²+2p-15=0 (p-3)(p+5)=0 so that we have +2p, and yet -15.

34. SolomonZelman

now p=?

p = 3 and -5

or, sorry I meant or.

37. SolomonZelman

yes p=3, p=-5.

38. SolomonZelman

this is the original prob, correct?

Yes.

40. SolomonZelman

yw