(2x+1)(3x-4)-x(2x+3)=1

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Solution for (2x+1)(3x-4)-x(2x+3)=1 equation: 


Simplifying
(2x + 1)(3x + -4) + -1x(2x + 3) = 1

Reorder the terms:
(1 + 2x)(3x + -4) + -1x(2x + 3) = 1

Reorder the terms:
(1 + 2x)(-4 + 3x) + -1x(2x + 3) = 1

Multiply (1 + 2x) * (-4 + 3x)
(1(-4 + 3x) + 2x * (-4 + 3x)) + -1x(2x + 3) = 1
((-4 * 1 + 3x * 1) + 2x * (-4 + 3x)) + -1x(2x + 3) = 1
((-4 + 3x) + 2x * (-4 + 3x)) + -1x(2x + 3) = 1
(-4 + 3x + (-4 * 2x + 3x * 2x)) + -1x(2x + 3) = 1
(-4 + 3x + (-8x + 6x2)) + -1x(2x + 3) = 1

Combine like terms: 3x + -8x = -5x
(-4 + -5x + 6x2) + -1x(2x + 3) = 1

Reorder the terms:
-4 + -5x + 6x2 + -1x(3 + 2x) = 1
-4 + -5x + 6x2 + (3 * -1x + 2x * -1x) = 1
-4 + -5x + 6x2 + (-3x + -2x2) = 1

Reorder the terms:
-4 + -5x + -3x + 6x2 + -2x2 = 1

Combine like terms: -5x + -3x = -8x
-4 + -8x + 6x2 + -2x2 = 1

Combine like terms: 6x2 + -2x2 = 4x2
-4 + -8x + 4x2 = 1

Solving
-4 + -8x + 4x2 = 1

Solving for variable 'x'.

Reorder the terms:
-4 + -1 + -8x + 4x2 = 1 + -1

Combine like terms: -4 + -1 = -5
-5 + -8x + 4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-5 + -8x + 4x2 = 0

Factor a trinomial.
(-1 + -2x)(5 + -2x) = 0


Subproblem 1
Set the factor '(-1 + -2x)' equal to zero and attempt to solve:

Simplifying
-1 + -2x = 0

Solving
-1 + -2x = 0

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

Add '1' to each side of the equation.
-1 + 1 + -2x = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2x = 0 + 1
-2x = 0 + 1

Combine like terms: 0 + 1 = 1
-2x = 1

Divide each side by '-2'.
x = -0.5

Simplifying
x = -0.5

Subproblem 2
Set the factor '(5 + -2x)' equal to zero and attempt to solve:

Simplifying
5 + -2x = 0

Solving
5 + -2x = 0

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

Add '-5' to each side of the equation.
5 + -5 + -2x = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -2x = 0 + -5
-2x = 0 + -5

Combine like terms: 0 + -5 = -5
-2x = -5

Divide each side by '-2'.
x = 2.5

Simplifying
x = 2.5
Solution
x = {-0.5, 2.5}

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