78=2x(x+3)

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Solution for 78=2x(x+3) equation: 


Simplifying
78 = 2x(x + 3)

Reorder the terms:
78 = 2x(3 + x)
78 = (3 * 2x + x * 2x)
78 = (6x + 2x2)

Solving
78 = 6x + 2x2

Solving for variable 'x'.

Reorder the terms:
78 + -6x + -2x2 = 6x + -6x + 2x2 + -2x2

Combine like terms: 6x + -6x = 0
78 + -6x + -2x2 = 0 + 2x2 + -2x2
78 + -6x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
78 + -6x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(39 + -3x + -1x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(39 + -3x + -1x2)' equal to zero and attempt to solve:

Simplifying
39 + -3x + -1x2 = 0

Solving
39 + -3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-39 + 3x + x2 = 0

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + 3x + 39 + x2 = 0 + 39

Reorder the terms:
-39 + 39 + 3x + x2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + 3x + x2 = 0 + 39
3x + x2 = 0 + 39

Combine like terms: 0 + 39 = 39
3x + x2 = 39

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 = 39 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 39 + 2.25

Combine like terms: 39 + 2.25 = 41.25
2.25 + 3x + x2 = 41.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 41.25

Calculate the square root of the right side: 6.422616289

Break this problem into two subproblems by setting 
(x + 1.5) equal to 6.422616289 and -6.422616289.


Subproblem 1
x + 1.5 = 6.422616289

Simplifying
x + 1.5 = 6.422616289

Reorder the terms:
1.5 + x = 6.422616289

Solving
1.5 + x = 6.422616289

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 = 6.422616289 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 6.422616289 + -1.5
x = 6.422616289 + -1.5

Combine like terms: 6.422616289 + -1.5 = 4.922616289
x = 4.922616289

Simplifying
x = 4.922616289


Subproblem 2
x + 1.5 = -6.422616289

Simplifying
x + 1.5 = -6.422616289

Reorder the terms:
1.5 + x = -6.422616289

Solving
1.5 + x = -6.422616289

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 = -6.422616289 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -6.422616289 + -1.5
x = -6.422616289 + -1.5

Combine like terms: -6.422616289 + -1.5 = -7.922616289
x = -7.922616289

Simplifying
x = -7.922616289


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.922616289, -7.922616289}
Solution
x = {4.922616289, -7.922616289}

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