4x^2-5x+3=(2x-1)(-x+7)

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Solution for 4x^2-5x+3=(2x-1)(-x+7) equation: 


Simplifying
4x2 + -5x + 3 = (2x + -1)(-1x + 7)

Reorder the terms:
3 + -5x + 4x2 = (2x + -1)(-1x + 7)

Reorder the terms:
3 + -5x + 4x2 = (-1 + 2x)(-1x + 7)

Reorder the terms:
3 + -5x + 4x2 = (-1 + 2x)(7 + -1x)

Multiply (-1 + 2x) * (7 + -1x)
3 + -5x + 4x2 = (-1(7 + -1x) + 2x * (7 + -1x))
3 + -5x + 4x2 = ((7 * -1 + -1x * -1) + 2x * (7 + -1x))
3 + -5x + 4x2 = ((-7 + 1x) + 2x * (7 + -1x))
3 + -5x + 4x2 = (-7 + 1x + (7 * 2x + -1x * 2x))
3 + -5x + 4x2 = (-7 + 1x + (14x + -2x2))

Combine like terms: 1x + 14x = 15x
3 + -5x + 4x2 = (-7 + 15x + -2x2)

Solving
3 + -5x + 4x2 = -7 + 15x + -2x2

Solving for variable 'x'.

Reorder the terms:
3 + 7 + -5x + -15x + 4x2 + 2x2 = -7 + 15x + -2x2 + 7 + -15x + 2x2

Combine like terms: 3 + 7 = 10
10 + -5x + -15x + 4x2 + 2x2 = -7 + 15x + -2x2 + 7 + -15x + 2x2

Combine like terms: -5x + -15x = -20x
10 + -20x + 4x2 + 2x2 = -7 + 15x + -2x2 + 7 + -15x + 2x2

Combine like terms: 4x2 + 2x2 = 6x2
10 + -20x + 6x2 = -7 + 15x + -2x2 + 7 + -15x + 2x2

Reorder the terms:
10 + -20x + 6x2 = -7 + 7 + 15x + -15x + -2x2 + 2x2

Combine like terms: -7 + 7 = 0
10 + -20x + 6x2 = 0 + 15x + -15x + -2x2 + 2x2
10 + -20x + 6x2 = 15x + -15x + -2x2 + 2x2

Combine like terms: 15x + -15x = 0
10 + -20x + 6x2 = 0 + -2x2 + 2x2
10 + -20x + 6x2 = -2x2 + 2x2

Combine like terms: -2x2 + 2x2 = 0
10 + -20x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + -10x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(5 + -10x + 3x2)' equal to zero and attempt to solve:

Simplifying
5 + -10x + 3x2 = 0

Solving
5 + -10x + 3x2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
1.666666667 + -3.333333333x + x2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + -3.333333333x + -1.666666667 + x2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + -3.333333333x + x2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + -3.333333333x + x2 = 0 + -1.666666667
-3.333333333x + x2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
-3.333333333x + x2 = -1.666666667

The x term is -3.333333333x.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333x + 2.777777779 + x2 = -1.666666667 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333x + x2 = -1.666666667 + 2.777777779

Combine like terms: -1.666666667 + 2.777777779 = 1.111111112
2.777777779 + -3.333333333x + x2 = 1.111111112

Factor a perfect square on the left side:
(x + -1.666666667)(x + -1.666666667) = 1.111111112

Calculate the square root of the right side: 1.054092554

Break this problem into two subproblems by setting 
(x + -1.666666667) equal to 1.054092554 and -1.054092554.


Subproblem 1
x + -1.666666667 = 1.054092554

Simplifying
x + -1.666666667 = 1.054092554

Reorder the terms:
-1.666666667 + x = 1.054092554

Solving
-1.666666667 + x = 1.054092554

Solving for variable 'x'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = 1.054092554 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = 1.054092554 + 1.666666667
x = 1.054092554 + 1.666666667

Combine like terms: 1.054092554 + 1.666666667 = 2.720759221
x = 2.720759221

Simplifying
x = 2.720759221


Subproblem 2
x + -1.666666667 = -1.054092554

Simplifying
x + -1.666666667 = -1.054092554

Reorder the terms:
-1.666666667 + x = -1.054092554

Solving
-1.666666667 + x = -1.054092554

Solving for variable 'x'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = -1.054092554 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = -1.054092554 + 1.666666667
x = -1.054092554 + 1.666666667

Combine like terms: -1.054092554 + 1.666666667 = 0.612574113
x = 0.612574113

Simplifying
x = 0.612574113


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.720759221, 0.612574113}
Solution
x = {2.720759221, 0.612574113}

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