5*X^0+13*X^1+3*X^2=1*X^0+1*X^1

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Solution for 5*X^0+13*X^1+3*X^2=1*X^0+1*X^1 equation: 


Simplifying
5X0 + 13X + 3X2 = 1X0 + 1X

Solving
5 + 13X + 3X2 = 1 + 1X

Solving for variable 'X'.

Reorder the terms:
5 + -1 + 13X + -1X + 3X2 = 1 + 1X + -1 + -1X

Combine like terms: 5 + -1 = 4
4 + 13X + -1X + 3X2 = 1 + 1X + -1 + -1X

Combine like terms: 13X + -1X = 12X
4 + 12X + 3X2 = 1 + 1X + -1 + -1X

Reorder the terms:
4 + 12X + 3X2 = 1 + -1 + 1X + -1X

Combine like terms: 1 + -1 = 0
4 + 12X + 3X2 = 0 + 1X + -1X
4 + 12X + 3X2 = 1X + -1X

Combine like terms: 1X + -1X = 0
4 + 12X + 3X2 = 0

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

Divide each side by '3'.
1.333333333 + 4X + X2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + 4X + -1.333333333 + X2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + 4X + X2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + 4X + X2 = 0 + -1.333333333
4X + X2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
4X + X2 = -1.333333333

The X term is 4X.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4X + 4 + X2 = -1.333333333 + 4

Reorder the terms:
4 + 4X + X2 = -1.333333333 + 4

Combine like terms: -1.333333333 + 4 = 2.666666667
4 + 4X + X2 = 2.666666667

Factor a perfect square on the left side:
(X + 2)(X + 2) = 2.666666667

Calculate the square root of the right side: 1.632993162

Break this problem into two subproblems by setting 
(X + 2) equal to 1.632993162 and -1.632993162.


Subproblem 1
X + 2 = 1.632993162

Simplifying
X + 2 = 1.632993162

Reorder the terms:
2 + X = 1.632993162

Solving
2 + X = 1.632993162

Solving for variable 'X'.

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

Add '-2' to each side of the equation.
2 + -2 + X = 1.632993162 + -2

Combine like terms: 2 + -2 = 0
0 + X = 1.632993162 + -2
X = 1.632993162 + -2

Combine like terms: 1.632993162 + -2 = -0.367006838
X = -0.367006838

Simplifying
X = -0.367006838


Subproblem 2
X + 2 = -1.632993162

Simplifying
X + 2 = -1.632993162

Reorder the terms:
2 + X = -1.632993162

Solving
2 + X = -1.632993162

Solving for variable 'X'.

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

Add '-2' to each side of the equation.
2 + -2 + X = -1.632993162 + -2

Combine like terms: 2 + -2 = 0
0 + X = -1.632993162 + -2
X = -1.632993162 + -2

Combine like terms: -1.632993162 + -2 = -3.632993162
X = -3.632993162

Simplifying
X = -3.632993162


Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {-0.367006838, -3.632993162}

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