78=(4x+3)(3x)

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Solution for 78=(4x+3)(3x) equation: 


Simplifying
78 = (4x + 3)(3x)

Reorder the terms:
78 = (3 + 4x)(3x)

Remove parenthesis around (3x)
78 = (3 + 4x) * 3x

Reorder the terms for easier multiplication:
78 = 3x(3 + 4x)
78 = (3 * 3x + 4x * 3x)
78 = (9x + 12x2)

Solving
78 = 9x + 12x2

Solving for variable 'x'.

Reorder the terms:
78 + -9x + -12x2 = 9x + -9x + 12x2 + -12x2

Combine like terms: 9x + -9x = 0
78 + -9x + -12x2 = 0 + 12x2 + -12x2
78 + -9x + -12x2 = 12x2 + -12x2

Combine like terms: 12x2 + -12x2 = 0
78 + -9x + -12x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(26 + -3x + -4x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(26 + -3x + -4x2)' equal to zero and attempt to solve:

Simplifying
26 + -3x + -4x2 = 0

Solving
26 + -3x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-6.5 + 0.75x + x2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + 0.75x + 6.5 + x2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + 0.75x + x2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + 0.75x + x2 = 0 + 6.5
0.75x + x2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
0.75x + x2 = 6.5

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = 6.5 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 6.5 + 0.140625

Combine like terms: 6.5 + 0.140625 = 6.640625
0.140625 + 0.75x + x2 = 6.640625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 6.640625

Calculate the square root of the right side: 2.576941016

Break this problem into two subproblems by setting 
(x + 0.375) equal to 2.576941016 and -2.576941016.


Subproblem 1
x + 0.375 = 2.576941016

Simplifying
x + 0.375 = 2.576941016

Reorder the terms:
0.375 + x = 2.576941016

Solving
0.375 + x = 2.576941016

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = 2.576941016 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = 2.576941016 + -0.375
x = 2.576941016 + -0.375

Combine like terms: 2.576941016 + -0.375 = 2.201941016
x = 2.201941016

Simplifying
x = 2.201941016


Subproblem 2
x + 0.375 = -2.576941016

Simplifying
x + 0.375 = -2.576941016

Reorder the terms:
0.375 + x = -2.576941016

Solving
0.375 + x = -2.576941016

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = -2.576941016 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = -2.576941016 + -0.375
x = -2.576941016 + -0.375

Combine like terms: -2.576941016 + -0.375 = -2.951941016
x = -2.951941016

Simplifying
x = -2.951941016


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.201941016, -2.951941016}
Solution
x = {2.201941016, -2.951941016}

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