(2x-3)(4x-5)+(2x-3)=0

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Solution for (2x-3)(4x-5)+(2x-3)=0 equation: 


Simplifying
(2x + -3)(4x + -5) + (2x + -3) = 0

Reorder the terms:
(-3 + 2x)(4x + -5) + (2x + -3) = 0

Reorder the terms:
(-3 + 2x)(-5 + 4x) + (2x + -3) = 0

Multiply (-3 + 2x) * (-5 + 4x)
(-3(-5 + 4x) + 2x * (-5 + 4x)) + (2x + -3) = 0
((-5 * -3 + 4x * -3) + 2x * (-5 + 4x)) + (2x + -3) = 0
((15 + -12x) + 2x * (-5 + 4x)) + (2x + -3) = 0
(15 + -12x + (-5 * 2x + 4x * 2x)) + (2x + -3) = 0
(15 + -12x + (-10x + 8x2)) + (2x + -3) = 0

Combine like terms: -12x + -10x = -22x
(15 + -22x + 8x2) + (2x + -3) = 0

Reorder the terms:
15 + -22x + 8x2 + (-3 + 2x) = 0

Remove parenthesis around (-3 + 2x)
15 + -22x + 8x2 + -3 + 2x = 0

Reorder the terms:
15 + -3 + -22x + 2x + 8x2 = 0

Combine like terms: 15 + -3 = 12
12 + -22x + 2x + 8x2 = 0

Combine like terms: -22x + 2x = -20x
12 + -20x + 8x2 = 0

Solving
12 + -20x + 8x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(3 + -5x + 2x2) = 0

Factor a trinomial.
4((1 + -1x)(3 + -2x)) = 0

Ignore the factor 4.

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

Simplifying
1 + -1x = 0

Solving
1 + -1x = 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 + -1x = 0 + -1

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

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

Divide each side by '-1'.
x = 1

Simplifying
x = 1

Subproblem 2
Set the factor '(3 + -2x)' equal to zero and attempt to solve:

Simplifying
3 + -2x = 0

Solving
3 + -2x = 0

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

Add '-3' to each side of the equation.
3 + -3 + -2x = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -2x = 0 + -3
-2x = 0 + -3

Combine like terms: 0 + -3 = -3
-2x = -3

Divide each side by '-2'.
x = 1.5

Simplifying
x = 1.5
Solution
x = {1, 1.5}

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