3x^2+14x+6=0

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Solution for 3x^2+14x+6=0 equation:


Simplifying
3x2 + 14x + 6 = 0

Reorder the terms:
6 + 14x + 3x2 = 0

Solving
6 + 14x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
2 + 4.666666667x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 4.666666667x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + 4.666666667x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 4.666666667x + x2 = 0 + -2
4.666666667x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
4.666666667x + x2 = -2

The x term is 4.666666667x.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667x + 5.444444448 + x2 = -2 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667x + x2 = -2 + 5.444444448

Combine like terms: -2 + 5.444444448 = 3.444444448
5.444444448 + 4.666666667x + x2 = 3.444444448

Factor a perfect square on the left side:
(x + 2.333333334)(x + 2.333333334) = 3.444444448

Calculate the square root of the right side: 1.855921455

Break this problem into two subproblems by setting 
(x + 2.333333334) equal to 1.855921455 and -1.855921455.

Subproblem 1

x + 2.333333334 = 1.855921455 Simplifying x + 2.333333334 = 1.855921455 Reorder the terms: 2.333333334 + x = 1.855921455 Solving 2.333333334 + x = 1.855921455 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + x = 1.855921455 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + x = 1.855921455 + -2.333333334 x = 1.855921455 + -2.333333334 Combine like terms: 1.855921455 + -2.333333334 = -0.477411879 x = -0.477411879 Simplifying x = -0.477411879

Subproblem 2

x + 2.333333334 = -1.855921455 Simplifying x + 2.333333334 = -1.855921455 Reorder the terms: 2.333333334 + x = -1.855921455 Solving 2.333333334 + x = -1.855921455 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + x = -1.855921455 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + x = -1.855921455 + -2.333333334 x = -1.855921455 + -2.333333334 Combine like terms: -1.855921455 + -2.333333334 = -4.189254789 x = -4.189254789 Simplifying x = -4.189254789

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.477411879, -4.189254789}

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