100-2x=3x^2-12x+14

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Solution for 100-2x=3x^2-12x+14 equation:


Simplifying
100 + -2x = 3x2 + -12x + 14

Reorder the terms:
100 + -2x = 14 + -12x + 3x2

Solving
100 + -2x = 14 + -12x + 3x2

Solving for variable 'x'.

Reorder the terms:
100 + -14 + -2x + 12x + -3x2 = 14 + -12x + 3x2 + -14 + 12x + -3x2

Combine like terms: 100 + -14 = 86
86 + -2x + 12x + -3x2 = 14 + -12x + 3x2 + -14 + 12x + -3x2

Combine like terms: -2x + 12x = 10x
86 + 10x + -3x2 = 14 + -12x + 3x2 + -14 + 12x + -3x2

Reorder the terms:
86 + 10x + -3x2 = 14 + -14 + -12x + 12x + 3x2 + -3x2

Combine like terms: 14 + -14 = 0
86 + 10x + -3x2 = 0 + -12x + 12x + 3x2 + -3x2
86 + 10x + -3x2 = -12x + 12x + 3x2 + -3x2

Combine like terms: -12x + 12x = 0
86 + 10x + -3x2 = 0 + 3x2 + -3x2
86 + 10x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
86 + 10x + -3x2 = 0

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

Divide each side by '-3'.
-28.66666667 + -3.333333333x + x2 = 0

Move the constant term to the right:

Add '28.66666667' to each side of the equation.
-28.66666667 + -3.333333333x + 28.66666667 + x2 = 0 + 28.66666667

Reorder the terms:
-28.66666667 + 28.66666667 + -3.333333333x + x2 = 0 + 28.66666667

Combine like terms: -28.66666667 + 28.66666667 = 0.00000000
0.00000000 + -3.333333333x + x2 = 0 + 28.66666667
-3.333333333x + x2 = 0 + 28.66666667

Combine like terms: 0 + 28.66666667 = 28.66666667
-3.333333333x + x2 = 28.66666667

The x term is -3.333333333x.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333x + 2.777777779 + x2 = 28.66666667 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333x + x2 = 28.66666667 + 2.777777779

Combine like terms: 28.66666667 + 2.777777779 = 31.444444449
2.777777779 + -3.333333333x + x2 = 31.444444449

Factor a perfect square on the left side:
(x + -1.666666667)(x + -1.666666667) = 31.444444449

Calculate the square root of the right side: 5.607534614

Break this problem into two subproblems by setting 
(x + -1.666666667) equal to 5.607534614 and -5.607534614.

Subproblem 1

x + -1.666666667 = 5.607534614 Simplifying x + -1.666666667 = 5.607534614 Reorder the terms: -1.666666667 + x = 5.607534614 Solving -1.666666667 + x = 5.607534614 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = 5.607534614 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = 5.607534614 + 1.666666667 x = 5.607534614 + 1.666666667 Combine like terms: 5.607534614 + 1.666666667 = 7.274201281 x = 7.274201281 Simplifying x = 7.274201281

Subproblem 2

x + -1.666666667 = -5.607534614 Simplifying x + -1.666666667 = -5.607534614 Reorder the terms: -1.666666667 + x = -5.607534614 Solving -1.666666667 + x = -5.607534614 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = -5.607534614 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = -5.607534614 + 1.666666667 x = -5.607534614 + 1.666666667 Combine like terms: -5.607534614 + 1.666666667 = -3.940867947 x = -3.940867947 Simplifying x = -3.940867947

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.274201281, -3.940867947}

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