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Apr 19, 2019
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Solve for x over the real numbers:
abs(2 x^2 + 3 x - 1)^(3 x^2 - 13 x + 4) = 1

Take the natural logarithm of both sides:
log(abs(2 x^2 + 3 x - 1)) (3 x^2 - 13 x + 4) = 0

The left hand side factors into a product with three terms:
log(abs(2 x^2 + 3 x - 1)) (x - 4) (3 x - 1) = 0

Split (x - 4) (3 x - 1) log(abs(2 x^2 + 3 x - 1)) into separate parts with additional assumptions.
Assume abs(2 x^2 + 3 x - 1)!=0 from log(abs(2 x^2 + 3 x - 1)):
x - 4 = 0 for abs(2 x^2 + 3 x - 1)!=0
 or 3 x - 1 = 0 for abs(2 x^2 + 3 x - 1)!=0
 or log(abs(2 x^2 + 3 x - 1)) = 0

Add 4 to both sides:
x = 4 or 3 x - 1 = 0 for abs(2 x^2 + 3 x - 1)!=0
 or log(abs(2 x^2 + 3 x - 1)) = 0

Add 1 to both sides:
x = 4 or 3 x = 1 for abs(2 x^2 + 3 x - 1)!=0
 or log(abs(2 x^2 + 3 x - 1)) = 0

Divide both sides by 3:
x = 4 or x = 1/3 or log(abs(2 x^2 + 3 x - 1)) = 0
Cancel logarithms by taking exp of both sides:
x = 4 or x = 1/3 or abs(2 x^2 + 3 x - 1) = 1

Split the equation into two possible cases:
x = 4 or x = 1/3 or 2 x^2 + 3 x - 1 = 1 or 2 x^2 + 3 x - 1 = -1

Divide both sides by 2:
x = 4 or x = 1/3 or x^2 + (3 x)/2 - 1/2 = 1/2 or 2 x^2 + 3 x - 1 = -1

Add 1/2 to both sides:
x = 4 or x = 1/3 or x^2 + (3 x)/2 = 1 or 2 x^2 + 3 x - 1 = -1

Add 9/16 to both sides:
x = 4 or x = 1/3 or x^2 + (3 x)/2 + 9/16 = 25/16 or 2 x^2 + 3 x - 1 = -1

Write the left hand side as a square:
x = 4 or x = 1/3 or (x + 3/4)^2 = 25/16 or 2 x^2 + 3 x - 1 = -1

Take the square root of both sides:
x = 4 or x = 1/3 or x + 3/4 = 5/4 or x + 3/4 = -5/4 or 2 x^2 + 3 x - 1 = -1

Subtract 3/4 from both sides:
x = 4 or x = 1/3 or x = 1/2 or x + 3/4 = -5/4 or 2 x^2 + 3 x - 1 = -1

Subtract 3/4 from both sides:
x = 4 or x = 1/3 or x = 1/2 or x = -2 or 2 x^2 + 3 x - 1 = -1

Divide both sides by 2:
x = 4 or x = 1/3 or x = 1/2 or x = -2 or x^2 + (3 x)/2 - 1/2 = -1/2

Add 1/2 to both sides:
x = 4 or x = 1/3 or x = 1/2 or x = -2 or x^2 + (3 x)/2 = 0

Factor x and constant terms from the left hand side:
x = 4 or x = 1/3 or x = 1/2 or x = -2 or 1/2 x (2 x + 3) = 0

Multiply both sides by 2:
x = 4 or x = 1/3 or x = 1/2 or x = -2 or x (2 x + 3) = 0

Split into two equations:
x = 4 or x = 1/3 or x = 1/2 or x = -2 or x = 0 or 2 x + 3 = 0
Subtract 3 from both sides:
x = 4 or x = 1/3 or x = 1/2 or x = -2 or x = 0 or 2 x = -3

Divide both sides by 2:
x = 4     or x = 1/3     or x = 1/2     or x = -2     or x = 0     or x = -3/2

Apr 19, 2019

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