SOLUTION: Solve the following equation by first doing a substitution to make it a quadratic. 3x^-2 + x^-1 -24=0

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Question 1061504: Solve the following equation by first doing a substitution to make it a quadratic.
3x^-2 + x^-1 -24=0
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E%28-1%29=1%2Fx
-
x%5E%28-2%29=1%2Fx%5E2


p=1%2Fx

3p%5E2%2Bp-24=0

Try to factorize.
%283p-8%29%28p%2B3%29=0

3p=8
p=8%2F3
OR
p=-3


system%281%2Fx=8%2F3%2COR%2C1%2Fx=-3%29


highlight%28system%28x=3%2F8%2COR%2Cx=-1%2F3%29%29

Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the following equation by first doing a substitution to make it a quadratic.
3x^-2 + x^-1 -24=0
3x%5E%28-+2%29+%2B+x%5E%28-+1%29+-+24+=+0

The SIMPLE substitution: Let matrix%281%2C3%2C+a%2C+be%2C+x%5E%28-+1%29%29

Then matrix%281%2C3%2C+3x%5E%28-+2%29%2C+becomes%2C+3a%5E2%29, and 3x%5E%28-+2%29+%2B+x%5E%28-+1%29+-+24+=+0 becomes: 3a%5E2+%2B+a+-+24+=+0

3a%5E2+%2B+9a+-+8a+-+24+=+0 ------ Repalacing a, with 9a - 8a

3a(a + 3) - 8(a + 3) = 0

(a + 3)(3a - 8) = 0

matrix%281%2C7%2C+a%2C+%22=%22%2C+-+3%2C+or%2C+a%2C+%22=%22%2C+8%2F3%29 



x%5E%28-+1%29+=+-+3 ------- Substituting back matrix%281%2C5%2C+a%2C+or%2C+-+3%2C+for%2C+x%5E%28-+1%29%29

x%5E%28%28-+1%29+%2A+%28-+1%29%29+=+%28-+3%29%5E%28-+1%29 ------ Raising each side to the - 1 power

highlight_green%28x+=+-+1%2F3%29 

Substitute this value for x (-+1%2F3) into the original equation. If this value satisfies the original equation, then x+=+-+1%2F3 is a solution to the equation. 

This value for x DOES SATISFY the equation.



x%5E%28-+1%29+=+%28-+8%29%2F3 ------- Substituting back matrix%281%2C5%2C+a%2C+or%2C+8%2F3%2C+for%2C+x%5E%28-+1%29%29

x%5E%28%28-+1%29+%2A+%28-+1%29%29+=+%288%2F3%29%5E%28-+1%29 ------ Raising each side to the - 1 power

highlight_green%28x+=+3%2F8%29 

Substitute this value for x (3%2F8) into the original equation. If this value satisfies the original equation, then x+=+3%2F8 is a solution to the equation.

This value for x DOES SATISFY the equation.