(.5)x(3x+5)=69

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Solution for (.5)x(3x+5)=69 equation: 


Simplifying
(0.5) * x(3x + 5) = 69

Reorder the terms:
0.5x(5 + 3x) = 69
(5 * 0.5x + 3x * 0.5x) = 69
(2.5x + 1.5x2) = 69

Solving
2.5x + 1.5x2 = 69

Solving for variable 'x'.

Reorder the terms:
-69 + 2.5x + 1.5x2 = 69 + -69

Combine like terms: 69 + -69 = 0
-69 + 2.5x + 1.5x2 = 0

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

Divide each side by '1.5'.
-46 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '46' to each side of the equation.
-46 + 1.666666667x + 46 + x2 = 0 + 46

Reorder the terms:
-46 + 46 + 1.666666667x + x2 = 0 + 46

Combine like terms: -46 + 46 = 0
0 + 1.666666667x + x2 = 0 + 46
1.666666667x + x2 = 0 + 46

Combine like terms: 0 + 46 = 46
1.666666667x + x2 = 46

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = 46 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = 46 + 0.6944444447

Combine like terms: 46 + 0.6944444447 = 46.6944444447
0.6944444447 + 1.666666667x + x2 = 46.6944444447

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 46.6944444447

Calculate the square root of the right side: 6.833333333

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 6.833333333 and -6.833333333.


Subproblem 1
x + 0.8333333335 = 6.833333333

Simplifying
x + 0.8333333335 = 6.833333333

Reorder the terms:
0.8333333335 + x = 6.833333333

Solving
0.8333333335 + x = 6.833333333

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = 6.833333333 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = 6.833333333 + -0.8333333335
x = 6.833333333 + -0.8333333335

Combine like terms: 6.833333333 + -0.8333333335 = 5.9999999995
x = 5.9999999995

Simplifying
x = 5.9999999995


Subproblem 2
x + 0.8333333335 = -6.833333333

Simplifying
x + 0.8333333335 = -6.833333333

Reorder the terms:
0.8333333335 + x = -6.833333333

Solving
0.8333333335 + x = -6.833333333

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = -6.833333333 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = -6.833333333 + -0.8333333335
x = -6.833333333 + -0.8333333335

Combine like terms: -6.833333333 + -0.8333333335 = -7.6666666665
x = -7.6666666665

Simplifying
x = -7.6666666665


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.9999999995, -7.6666666665}

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