6x^2+16x+3=0

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Solution for 6x^2+16x+3=0 equation:


Simplifying
6x2 + 16x + 3 = 0

Reorder the terms:
3 + 16x + 6x2 = 0

Solving
3 + 16x + 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'.
0.5 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2.666666667x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2.666666667x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2.666666667x + x2 = 0 + -0.5
2.666666667x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2.666666667x + x2 = -0.5

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 = -0.5 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = -0.5 + 1.777777780

Combine like terms: -0.5 + 1.777777780 = 1.27777778
1.777777780 + 2.666666667x + x2 = 1.27777778

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 1.27777778

Calculate the square root of the right side: 1.130388332

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 1.130388332 and -1.130388332.

Subproblem 1

x + 1.333333334 = 1.130388332 Simplifying x + 1.333333334 = 1.130388332 Reorder the terms: 1.333333334 + x = 1.130388332 Solving 1.333333334 + x = 1.130388332 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 = 1.130388332 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 1.130388332 + -1.333333334 x = 1.130388332 + -1.333333334 Combine like terms: 1.130388332 + -1.333333334 = -0.202945002 x = -0.202945002 Simplifying x = -0.202945002

Subproblem 2

x + 1.333333334 = -1.130388332 Simplifying x + 1.333333334 = -1.130388332 Reorder the terms: 1.333333334 + x = -1.130388332 Solving 1.333333334 + x = -1.130388332 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 = -1.130388332 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -1.130388332 + -1.333333334 x = -1.130388332 + -1.333333334 Combine like terms: -1.130388332 + -1.333333334 = -2.463721666 x = -2.463721666 Simplifying x = -2.463721666

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.202945002, -2.463721666}

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