3x^2+11x+9=0

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Solution for 3x^2+11x+9=0 equation:


Simplifying
3x2 + 11x + 9 = 0

Reorder the terms:
9 + 11x + 3x2 = 0

Solving
9 + 11x + 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'.
3 + 3.666666667x + x2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 3.666666667x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + 3.666666667x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 3.666666667x + x2 = 0 + -3
3.666666667x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
3.666666667x + x2 = -3

The x term is 3.666666667x.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667x + 3.361111114 + x2 = -3 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667x + x2 = -3 + 3.361111114

Combine like terms: -3 + 3.361111114 = 0.361111114
3.361111114 + 3.666666667x + x2 = 0.361111114

Factor a perfect square on the left side:
(x + 1.833333334)(x + 1.833333334) = 0.361111114

Calculate the square root of the right side: 0.600925215

Break this problem into two subproblems by setting 
(x + 1.833333334) equal to 0.600925215 and -0.600925215.

Subproblem 1

x + 1.833333334 = 0.600925215 Simplifying x + 1.833333334 = 0.600925215 Reorder the terms: 1.833333334 + x = 0.600925215 Solving 1.833333334 + x = 0.600925215 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = 0.600925215 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 0.600925215 + -1.833333334 x = 0.600925215 + -1.833333334 Combine like terms: 0.600925215 + -1.833333334 = -1.232408119 x = -1.232408119 Simplifying x = -1.232408119

Subproblem 2

x + 1.833333334 = -0.600925215 Simplifying x + 1.833333334 = -0.600925215 Reorder the terms: 1.833333334 + x = -0.600925215 Solving 1.833333334 + x = -0.600925215 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + x = -0.600925215 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -0.600925215 + -1.833333334 x = -0.600925215 + -1.833333334 Combine like terms: -0.600925215 + -1.833333334 = -2.434258549 x = -2.434258549 Simplifying x = -2.434258549

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.232408119, -2.434258549}

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