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

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


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

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

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

Multiply (-2 + 3x) * (3 + 4x)
(-2(3 + 4x) + 3x * (3 + 4x)) = 5x(x + -4)
((3 * -2 + 4x * -2) + 3x * (3 + 4x)) = 5x(x + -4)
((-6 + -8x) + 3x * (3 + 4x)) = 5x(x + -4)
(-6 + -8x + (3 * 3x + 4x * 3x)) = 5x(x + -4)
(-6 + -8x + (9x + 12x2)) = 5x(x + -4)

Combine like terms: -8x + 9x = 1x
(-6 + 1x + 12x2) = 5x(x + -4)

Reorder the terms:
-6 + 1x + 12x2 = 5x(-4 + x)
-6 + 1x + 12x2 = (-4 * 5x + x * 5x)
-6 + 1x + 12x2 = (-20x + 5x2)

Solving
-6 + 1x + 12x2 = -20x + 5x2

Solving for variable 'x'.

Reorder the terms:
-6 + 1x + 20x + 12x2 + -5x2 = -20x + 5x2 + 20x + -5x2

Combine like terms: 1x + 20x = 21x
-6 + 21x + 12x2 + -5x2 = -20x + 5x2 + 20x + -5x2

Combine like terms: 12x2 + -5x2 = 7x2
-6 + 21x + 7x2 = -20x + 5x2 + 20x + -5x2

Reorder the terms:
-6 + 21x + 7x2 = -20x + 20x + 5x2 + -5x2

Combine like terms: -20x + 20x = 0
-6 + 21x + 7x2 = 0 + 5x2 + -5x2
-6 + 21x + 7x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
-6 + 21x + 7x2 = 0

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

Divide each side by '7'.
-0.8571428571 + 3x + x2 = 0

Move the constant term to the right:

Add '0.8571428571' to each side of the equation.
-0.8571428571 + 3x + 0.8571428571 + x2 = 0 + 0.8571428571

Reorder the terms:
-0.8571428571 + 0.8571428571 + 3x + x2 = 0 + 0.8571428571

Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000
0.0000000000 + 3x + x2 = 0 + 0.8571428571
3x + x2 = 0 + 0.8571428571

Combine like terms: 0 + 0.8571428571 = 0.8571428571
3x + x2 = 0.8571428571

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 0.8571428571 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 0.8571428571 + 2.25

Combine like terms: 0.8571428571 + 2.25 = 3.1071428571
2.25 + 3x + x2 = 3.1071428571

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 3.1071428571

Calculate the square root of the right side: 1.762708954

Break this problem into two subproblems by setting 
(x + 1.5) equal to 1.762708954 and -1.762708954.

Subproblem 1

x + 1.5 = 1.762708954 Simplifying x + 1.5 = 1.762708954 Reorder the terms: 1.5 + x = 1.762708954 Solving 1.5 + x = 1.762708954 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.762708954 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.762708954 + -1.5 x = 1.762708954 + -1.5 Combine like terms: 1.762708954 + -1.5 = 0.262708954 x = 0.262708954 Simplifying x = 0.262708954

Subproblem 2

x + 1.5 = -1.762708954 Simplifying x + 1.5 = -1.762708954 Reorder the terms: 1.5 + x = -1.762708954 Solving 1.5 + x = -1.762708954 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.762708954 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.762708954 + -1.5 x = -1.762708954 + -1.5 Combine like terms: -1.762708954 + -1.5 = -3.262708954 x = -3.262708954 Simplifying x = -3.262708954

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.262708954, -3.262708954}

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