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What are the real solutions to the equation below?

(2x^2-5)

---------- ^ (x^2-3x)  -1=0                                  (Note: ----- is a fracion bar and the fraction is being raised to (x^2-3x) and 

     3                                                                   and then you take the aswer to the fraction raised to (x^2-3x) and -1=0)

 

 

     Also note that this is a quadratic...

 

Answer and Explination needed as soon as possible but will no be needed after tomorrow... Thank you!!!!

Trinityvamp286  Jul 9, 2017
edited by Guest  Jul 9, 2017
edited by Guest  Jul 9, 2017
edited by Guest  Jul 9, 2017
 #1
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I read your equation as: [(2x^2 - 5)/3]^(x^2 - 3x) - 1 = 0, solve for x


3^(3 x - x^2) (2 x^2 - 5)^(x^2 - 3 x) - 1 = 0

Divide both sides by 3^(3 x - x^2):
(2 x^2 - 5)^(x^2 - 3 x) - 3^(x^2 - 3 x) = 0

Add 3^(x^2 - 3 x) to both sides:
(2 x^2 - 5)^(x^2 - 3 x) = 3^(x^2 - 3 x)

Take the natural logarithm of both sides and use the identity log(a^b) = b log(a):
log(2 x^2 - 5) (x^2 - 3 x) = log(3) (x^2 - 3 x)

(x^2 - 3 x) log(2 x^2 - 5) = x^2 log(2 x^2 - 5) - 3 x log(2 x^2 - 5):
x^2 log(2 x^2 - 5) - 3 x log(2 x^2 - 5) = log(3) (x^2 - 3 x)

Expand out terms of the right hand side:
x^2 log(2 x^2 - 5) - 3 x log(2 x^2 - 5) = log(3) x^2 - 3 log(3) x

Subtract x^2 log(3) - 3 x log(3) from both sides:
3 log(3) x - log(3) x^2 - 3 x log(2 x^2 - 5) + x^2 log(2 x^2 - 5) = 0

The left hand side factors into a product with four terms:
-x (x - 3) (log(3) - log(2 x^2 - 5)) = 0

Multiply both sides by -1:
x (x - 3) (log(3) - log(2 x^2 - 5)) = 0

Split x (x - 3) (log(3) - log(2 x^2 - 5)) into separate parts with additional assumptions.
Assume 2 x^2 - 5!=0 from log(2 x^2 - 5):
x - 3 = 0 for 2 x^2 - 5!=0
 or x = 0 or log(3) - log(2 x^2 - 5) = 0

Add 3 to both sides:
x = 3 or x = 0 or log(3) - log(2 x^2 - 5) = 0

Subtract log(3) from both sides:
x = 3 or x = 0 or -log(2 x^2 - 5) = -log(3)

Multiply both sides by -1:
x = 3 or x = 0 or log(2 x^2 - 5) = log(3)

Cancel logarithms by taking exp of both sides:
x = 3 or x = 0 or 2 x^2 - 5 = 3

Add 5 to both sides:
x = 3 or x = 0 or 2 x^2 = 8

Divide both sides by 2:
x = 3 or x = 0 or x^2 = 4

Take the square root of both sides:
Answer: | x = 3   or   x = 0   or   x = 2   or    x = -2

Guest Jul 9, 2017

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