Non linear equation with negative exponents

[tklopfstein]

w‾² +5w‾¹-36=0

5w² (1__ + 5____ = 36)
w² W¹


5 + w = 180w²

180w² - w -5 =0

(36w +4)(5w-5)=0


Can’t figure out the factoring here?
I know that 36 and 5 have to be involved to equal 180. How do you know what to plug in for 2nd part. I get so confused.

Answer is -1/9 or 1/4

 

[tkhunny]

w‾² +5w‾¹-36=0
5w² (1__ + 5____ = 36)
w² W¹

You don't really show much concept of factoring. You may need to back up a bit. Switching things back and forth across the equal sign is not a good pattern. Let's do a free one and see if it starts to soak in a little.

This problem is a little tricky ONLY because of the negative exponents. Let's get rid of them by making a substitution. z = 1/w

Under this substitution w^-2 = z² and w^-1 = z¹

This makes the equation:

z² + 5*z - 36 = 0

We can now treat this like any other old quadratic, but we'll have to remember that we are working in 'z' and that won't give us the answer to the problem statement. We'll have to get back to 'w' before we are done.

z² + 5*z - 36 = 0

Set it up.

(z + ____)*(z - ____) = 0

All we need are factors of -36 that give +5. Hmmm... How about 9 * (-4), since 9-4 = 5!

(z + 9)*(z - 4) = 0

This leads to z = -9 and z = 4

Getting back to 'w' leads to w = -1/9 and w = 1/4

What do you think? Are you following?

P.S. I probably should mention that anyone whose screen name starts with 'tk' can't be all bad.

 

[Denis]

[quote="tkhunnyUnder this substitution w^-2 = z² and w^-1 = z²

go quick and edit that 2nd z^2, tk!

P.S. I probably should mention that anyone whose screen name starts with 'tk' can't be all bad.

...do you really ThinK so?
[/quote]

Tammy, in case you're still confused:
a^(-b) = 1 / a^b ; example:
3^(-2) = 1 / 3^2

 

[soroban]

Hello, tklopfstein!

w^-2 +5w^-1 - 36 = 0

Multiply through by w²: . 1 + 5w - 36w² .= .0

We have the quadratic: . 36w² - 5w - 1 .= .0

. . which factors: . (4w - 1)(9w + 1) .= .0

. . and has two roots:
. . . . . (a) . 4w - 1 = 0 . . .---> . . 4w = .1 . . ---> . . w = 1/4
. . . . . (b) . 9w + 1 = 0 . . ---> . . 9w = -1 . . ---> . . w = -1/9

(36w + 4)(5w - 5) = 0 . . . ??

This can't be right!

You'd have: . 4(9w + 1)·5(w - 1) . = . 20(9w + 1)(w - 1)
. . which means you could have factored out 20 in the very first step.

 

[tklopfstein]

Soroban,

I'm just wondering how can I get the (w-1) to equal 1/4. I think I'm not understanding something here. Which of course is not out of the ordinary for me.

Thanks so much for your help.

 

[tklopfstein]

Non Linear equation w/ neg exp

Dear tkhunny:

I think I finally figured it out. Sorry, it takes me awhile. (z+9) solution is -9 and convert it back to the original form of 1 over w leads to -1/9 and
then the other side is (z-4) solution is 4 and to convert it back to the orignal form of 1 over w it becomes 1/4.


Thanks so much,

 

[Denis]

Soroban,
I'm just wondering how can I get the (w-1) to equal 1/4. I think I'm not understanding something here. Which of course is not out of the ordinary for me. Thanks so much for your help.

Where do you get (w-1)? It's (4w-1) !
4w - 1 = 0
4w = 1
w = 1/4

you ok now?