## Destinyyyy one year ago Solve the equation by making an appropriate substitution.

1. anonymous

x^-2 +3x^-1 -10=0

2. freckles

let u=x^(-1) then u^2=x^(-2)

3. anonymous

okay

4. anonymous

Do i factor now?

5. freckles

yep yep

6. freckles

and you should be factoring u^2+3u-10 right now

7. anonymous

(u-2)(u+5)

8. freckles

neat so you can solve the equation (u-2)(u+5)=0 for u

9. anonymous

u=2, -5

10. freckles

now remember we replace x^(-1) with u so we have to undo that replacement by replacing u with x^(-1)

11. freckles

$x^{-1}=2 \text{ or } -5 \\ x^{-1}=2 \text{ or } x^{-1}=-5$

12. freckles

recall x^(-1) is the same thing as saying 1/x

13. freckles

$\frac{1}{x}=2 \text{ or } \frac{1}{x}=-5$

14. freckles

can you solve both of those equations for x

15. anonymous

?? x^-2=2 you have two -1

16. freckles

$\frac{1}{x}=a \\ \text{ multiply both sides by } x \\ 1=ax \\ \text{ divide both sides by } a \\ \frac{1}{a}=x \\ \text{ so assuming } x \neq 0 \text{ and } a \neq 0 \\ \text{ then } \frac{1}{x}=a \implies x=\frac{1}{a}$

17. anonymous

Um okay I think I understand that but dont see how that solves this.. :/

18. anonymous

Dont I just square root both sides?

19. freckles

you have 1/x=2 so x=1/2

20. freckles

you have 1/x=-5 so x=?

21. anonymous

-1/5

22. freckles

right

23. anonymous

So thats the answer?

24. freckles

x=1/2 or x=-1/5 yes

25. anonymous

Do I check it? Or no?

26. freckles

another way to solve this is multiply both sides of your equation by x^2 $x^{-2}+3x^{-1}-10=0 \\ x^2(x^{-2}+3x^{-1}-10)=x^{2}(0) \\ x^{2+(-2)}+3x^{2+(-1)}-10x^{2}=0 \\ x^{0}+3x^{1}-10x^2=0 \\ -10x^2+3x+1=0 \text{ since } x^{0}=1 \text{ and I just reorder terms } \\ -10x^2+5x-2x+1=0 \\ -5x(2x-1)-1(2x-1)=0 \\ (2x-1)(-5x-1)=0 \\ 2x-1=0 \text{ when } x=\frac{1}{2} \\ -5x-1=0 \text{ when } x=\frac{-1}{5}$ there is no harm in checking but the answers are fine we didn't raise both sides to an even power and are answers fit in with domain of the original equation (for this question we definitely did not want x=0 as a solution which we didn't have to worry about here)

27. freckles

oh I see what you did about you replace u with x^(-2) even though u equals x^(-1)

28. anonymous

Um okay

29. freckles

above not about :p

30. freckles

anyways if you have more questions post a new question I'm gonna take a short break

31. freckles

unless you have questions on this one then I can stay and answer them real quick

32. freckles

ok I will be back later! peace for now! :)

33. anonymous

Sorry didnt see you send back.