x(3x+8)=15

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Solution for x(3x+8)=15 equation:


Simplifying
x(3x + 8) = 15

Reorder the terms:
x(8 + 3x) = 15
(8 * x + 3x * x) = 15
(8x + 3x2) = 15

Solving
8x + 3x2 = 15

Solving for variable 'x'.

Reorder the terms:
-15 + 8x + 3x2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + 8x + 3x2 = 0

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

Divide each side by '3'.
-5 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 2.666666667x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + 2.666666667x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 2.666666667x + x2 = 0 + 5
2.666666667x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
2.666666667x + x2 = 5

The x term is 2.666666667x.  Take half its coefficient (1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
2.666666667x + 1.777777780 + x2 = 5 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = 5 + 1.777777780

Combine like terms: 5 + 1.777777780 = 6.77777778
1.777777780 + 2.666666667x + x2 = 6.77777778

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 6.77777778

Calculate the square root of the right side: 2.603416559

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 2.603416559 and -2.603416559.


Subproblem 1
x + 1.333333334 = 2.603416559

Simplifying
x + 1.333333334 = 2.603416559

Reorder the terms:
1.333333334 + x = 2.603416559

Solving
1.333333334 + x = 2.603416559

Solving for variable 'x'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + x = 2.603416559 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = 2.603416559 + -1.333333334
x = 2.603416559 + -1.333333334

Combine like terms: 2.603416559 + -1.333333334 = 1.270083225
x = 1.270083225

Simplifying
x = 1.270083225


Subproblem 2
x + 1.333333334 = -2.603416559

Simplifying
x + 1.333333334 = -2.603416559

Reorder the terms:
1.333333334 + x = -2.603416559

Solving
1.333333334 + x = -2.603416559

Solving for variable 'x'.

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

Add '-1.333333334' to each side of the equation.
1.333333334 + -1.333333334 + x = -2.603416559 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = -2.603416559 + -1.333333334
x = -2.603416559 + -1.333333334

Combine like terms: -2.603416559 + -1.333333334 = -3.936749893
x = -3.936749893

Simplifying
x = -3.936749893


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.270083225, -3.936749893}

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