3x^2-7x+15x=8

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Solution for 3x^2-7x+15x=8 equation: 


Simplifying
3x2 + -7x + 15x = 8

Reorder the terms:
-7x + 15x + 3x2 = 8

Combine like terms: -7x + 15x = 8x
8x + 3x2 = 8

Solving
8x + 3x2 = 8

Solving for variable 'x'.

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

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

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

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

Move the constant term to the right:

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

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

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + 2.666666667x + x2 = 0 + 2.666666667
2.666666667x + x2 = 0 + 2.666666667

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

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 = 2.666666667 + 1.777777780

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

Combine like terms: 2.666666667 + 1.777777780 = 4.444444447
1.777777780 + 2.666666667x + x2 = 4.444444447

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 4.444444447

Calculate the square root of the right side: 2.108185107

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 2.108185107 and -2.108185107.


Subproblem 1
x + 1.333333334 = 2.108185107

Simplifying
x + 1.333333334 = 2.108185107

Reorder the terms:
1.333333334 + x = 2.108185107

Solving
1.333333334 + x = 2.108185107

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 = 2.108185107 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = 2.108185107 + -1.333333334
x = 2.108185107 + -1.333333334

Combine like terms: 2.108185107 + -1.333333334 = 0.774851773
x = 0.774851773

Simplifying
x = 0.774851773


Subproblem 2
x + 1.333333334 = -2.108185107

Simplifying
x + 1.333333334 = -2.108185107

Reorder the terms:
1.333333334 + x = -2.108185107

Solving
1.333333334 + x = -2.108185107

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 = -2.108185107 + -1.333333334

Combine like terms: 1.333333334 + -1.333333334 = 0.000000000
0.000000000 + x = -2.108185107 + -1.333333334
x = -2.108185107 + -1.333333334

Combine like terms: -2.108185107 + -1.333333334 = -3.441518441
x = -3.441518441

Simplifying
x = -3.441518441


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.774851773, -3.441518441}

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