ln(3*x)-2*x^2+8x+16=ln(x+4)

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Solution for ln(3*x)-2*x^2+8x+16=ln(x+4) equation:


Simplifying
ln(3x) + -2x2 + 8x + 16 = ln(x + 4)

Remove parenthesis around (3x)
ln * 3x + -2x2 + 8x + 16 = ln(x + 4)

Reorder the terms for easier multiplication:
3ln * x + -2x2 + 8x + 16 = ln(x + 4)

Multiply ln * x
3lnx + -2x2 + 8x + 16 = ln(x + 4)

Reorder the terms:
16 + 3lnx + 8x + -2x2 = ln(x + 4)

Reorder the terms:
16 + 3lnx + 8x + -2x2 = ln(4 + x)
16 + 3lnx + 8x + -2x2 = (4 * ln + x * ln)
16 + 3lnx + 8x + -2x2 = (4ln + lnx)

Solving
16 + 3lnx + 8x + -2x2 = 4ln + lnx

Solving for variable 'l'.

Move all terms containing l to the left, all other terms to the right.

Add '-4ln' to each side of the equation.
16 + 3lnx + 8x + -4ln + -2x2 = 4ln + -4ln + lnx

Reorder the terms:
16 + -4ln + 3lnx + 8x + -2x2 = 4ln + -4ln + lnx

Combine like terms: 4ln + -4ln = 0
16 + -4ln + 3lnx + 8x + -2x2 = 0 + lnx
16 + -4ln + 3lnx + 8x + -2x2 = lnx

Add '-1lnx' to each side of the equation.
16 + -4ln + 3lnx + 8x + -1lnx + -2x2 = lnx + -1lnx

Reorder the terms:
16 + -4ln + 3lnx + -1lnx + 8x + -2x2 = lnx + -1lnx

Combine like terms: 3lnx + -1lnx = 2lnx
16 + -4ln + 2lnx + 8x + -2x2 = lnx + -1lnx

Combine like terms: lnx + -1lnx = 0
16 + -4ln + 2lnx + 8x + -2x2 = 0

Add '-16' to each side of the equation.
16 + -4ln + 2lnx + 8x + -16 + -2x2 = 0 + -16

Reorder the terms:
16 + -16 + -4ln + 2lnx + 8x + -2x2 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + -4ln + 2lnx + 8x + -2x2 = 0 + -16
-4ln + 2lnx + 8x + -2x2 = 0 + -16

Combine like terms: 0 + -16 = -16
-4ln + 2lnx + 8x + -2x2 = -16

Add '-8x' to each side of the equation.
-4ln + 2lnx + 8x + -8x + -2x2 = -16 + -8x

Combine like terms: 8x + -8x = 0
-4ln + 2lnx + 0 + -2x2 = -16 + -8x
-4ln + 2lnx + -2x2 = -16 + -8x

Add '2x2' to each side of the equation.
-4ln + 2lnx + -2x2 + 2x2 = -16 + -8x + 2x2

Combine like terms: -2x2 + 2x2 = 0
-4ln + 2lnx + 0 = -16 + -8x + 2x2
-4ln + 2lnx = -16 + -8x + 2x2

Reorder the terms:
16 + -4ln + 2lnx + 8x + -2x2 = -16 + -8x + 2x2 + 16 + 8x + -2x2

Reorder the terms:
16 + -4ln + 2lnx + 8x + -2x2 = -16 + 16 + -8x + 8x + 2x2 + -2x2

Combine like terms: -16 + 16 = 0
16 + -4ln + 2lnx + 8x + -2x2 = 0 + -8x + 8x + 2x2 + -2x2
16 + -4ln + 2lnx + 8x + -2x2 = -8x + 8x + 2x2 + -2x2

Combine like terms: -8x + 8x = 0
16 + -4ln + 2lnx + 8x + -2x2 = 0 + 2x2 + -2x2
16 + -4ln + 2lnx + 8x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
16 + -4ln + 2lnx + 8x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(8 + -2ln + lnx + 4x + -1x2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(8 + -2ln + lnx + 4x + -1x2)' equal to zero and attempt to solve: Simplifying 8 + -2ln + lnx + 4x + -1x2 = 0 Solving 8 + -2ln + lnx + 4x + -1x2 = 0 Move all terms containing l to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -2ln + lnx + 4x + -8 + -1x2 = 0 + -8 Reorder the terms: 8 + -8 + -2ln + lnx + 4x + -1x2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + -2ln + lnx + 4x + -1x2 = 0 + -8 -2ln + lnx + 4x + -1x2 = 0 + -8 Combine like terms: 0 + -8 = -8 -2ln + lnx + 4x + -1x2 = -8 Add '-4x' to each side of the equation. -2ln + lnx + 4x + -4x + -1x2 = -8 + -4x Combine like terms: 4x + -4x = 0 -2ln + lnx + 0 + -1x2 = -8 + -4x -2ln + lnx + -1x2 = -8 + -4x Add 'x2' to each side of the equation. -2ln + lnx + -1x2 + x2 = -8 + -4x + x2 Combine like terms: -1x2 + x2 = 0 -2ln + lnx + 0 = -8 + -4x + x2 -2ln + lnx = -8 + -4x + x2 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined.

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