6x^2+2x-21=0

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Solution for 6x^2+2x-21=0 equation:


Simplifying
6x2 + 2x + -21 = 0

Reorder the terms:
-21 + 2x + 6x2 = 0

Solving
-21 + 2x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-3.5 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 0.3333333333x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 0.3333333333x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 0.3333333333x + x2 = 0 + 3.5
0.3333333333x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
0.3333333333x + x2 = 3.5

The x term is 0.3333333333x.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333x + 0.02777777779 + x2 = 3.5 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x + x2 = 3.5 + 0.02777777779

Combine like terms: 3.5 + 0.02777777779 = 3.52777777779
0.02777777779 + 0.3333333333x + x2 = 3.52777777779

Factor a perfect square on the left side:
(x + 0.1666666667)(x + 0.1666666667) = 3.52777777779

Calculate the square root of the right side: 1.878237945

Break this problem into two subproblems by setting 
(x + 0.1666666667) equal to 1.878237945 and -1.878237945.

Subproblem 1

x + 0.1666666667 = 1.878237945 Simplifying x + 0.1666666667 = 1.878237945 Reorder the terms: 0.1666666667 + x = 1.878237945 Solving 0.1666666667 + x = 1.878237945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 1.878237945 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 1.878237945 + -0.1666666667 x = 1.878237945 + -0.1666666667 Combine like terms: 1.878237945 + -0.1666666667 = 1.7115712783 x = 1.7115712783 Simplifying x = 1.7115712783

Subproblem 2

x + 0.1666666667 = -1.878237945 Simplifying x + 0.1666666667 = -1.878237945 Reorder the terms: 0.1666666667 + x = -1.878237945 Solving 0.1666666667 + x = -1.878237945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -1.878237945 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -1.878237945 + -0.1666666667 x = -1.878237945 + -0.1666666667 Combine like terms: -1.878237945 + -0.1666666667 = -2.0449046117 x = -2.0449046117 Simplifying x = -2.0449046117

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.7115712783, -2.0449046117}

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