6x^2+16x+9=0

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Solution for 6x^2+16x+9=0 equation:


Simplifying
6x2 + 16x + 9 = 0

Reorder the terms:
9 + 16x + 6x2 = 0

Solving
9 + 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'.
1.5 + 2.666666667x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 2.666666667x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 2.666666667x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 2.666666667x + x2 = 0 + -1.5
2.666666667x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
2.666666667x + x2 = -1.5

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 = -1.5 + 1.777777780

Reorder the terms:
1.777777780 + 2.666666667x + x2 = -1.5 + 1.777777780

Combine like terms: -1.5 + 1.777777780 = 0.27777778
1.777777780 + 2.666666667x + x2 = 0.27777778

Factor a perfect square on the left side:
(x + 1.333333334)(x + 1.333333334) = 0.27777778

Calculate the square root of the right side: 0.527046279

Break this problem into two subproblems by setting 
(x + 1.333333334) equal to 0.527046279 and -0.527046279.

Subproblem 1

x + 1.333333334 = 0.527046279 Simplifying x + 1.333333334 = 0.527046279 Reorder the terms: 1.333333334 + x = 0.527046279 Solving 1.333333334 + x = 0.527046279 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 = 0.527046279 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 0.527046279 + -1.333333334 x = 0.527046279 + -1.333333334 Combine like terms: 0.527046279 + -1.333333334 = -0.806287055 x = -0.806287055 Simplifying x = -0.806287055

Subproblem 2

x + 1.333333334 = -0.527046279 Simplifying x + 1.333333334 = -0.527046279 Reorder the terms: 1.333333334 + x = -0.527046279 Solving 1.333333334 + x = -0.527046279 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 = -0.527046279 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -0.527046279 + -1.333333334 x = -0.527046279 + -1.333333334 Combine like terms: -0.527046279 + -1.333333334 = -1.860379613 x = -1.860379613 Simplifying x = -1.860379613

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.806287055, -1.860379613}

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