Question

Solve the equation by making an appropriate substitution.

Answer 1

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

Answer 2

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

Answer 3

okay

Answer 4

Do i factor now?

Answer 5

yep yep

Answer 6

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

Answer 7

(u-2)(u+5)

Answer 8

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

Answer 9

u=2, -5

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

can you solve both of those equations for x

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

Dont I just square root both sides?

Answer 19

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

Answer 20

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

Answer 21

-1/5

Answer 22

right

Answer 23

So thats the answer?

Answer 24

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

Answer 25

Do I check it? Or no?

Answer 26

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)

Answer 27

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

Answer 28

Um okay

Answer 29

above not about :p

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

Sorry didnt see you send back.