3x^2+9x+1=0

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Solution for 3x^2+9x+1=0 equation:


Simplifying
3x2 + 9x + 1 = 0

Reorder the terms:
1 + 9x + 3x2 = 0

Solving
1 + 9x + 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'.
0.3333333333 + 3x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 3x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 3x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 3x + x2 = 0 + -0.3333333333
3x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
3x + x2 = -0.3333333333

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = -0.3333333333 + 2.25

Reorder the terms:
2.25 + 3x + x2 = -0.3333333333 + 2.25

Combine like terms: -0.3333333333 + 2.25 = 1.9166666667
2.25 + 3x + x2 = 1.9166666667

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 1.9166666667

Calculate the square root of the right side: 1.38443731

Break this problem into two subproblems by setting 
(x + 1.5) equal to 1.38443731 and -1.38443731.

Subproblem 1

x + 1.5 = 1.38443731 Simplifying x + 1.5 = 1.38443731 Reorder the terms: 1.5 + x = 1.38443731 Solving 1.5 + x = 1.38443731 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.38443731 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.38443731 + -1.5 x = 1.38443731 + -1.5 Combine like terms: 1.38443731 + -1.5 = -0.11556269 x = -0.11556269 Simplifying x = -0.11556269

Subproblem 2

x + 1.5 = -1.38443731 Simplifying x + 1.5 = -1.38443731 Reorder the terms: 1.5 + x = -1.38443731 Solving 1.5 + x = -1.38443731 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.38443731 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.38443731 + -1.5 x = -1.38443731 + -1.5 Combine like terms: -1.38443731 + -1.5 = -2.88443731 x = -2.88443731 Simplifying x = -2.88443731

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.11556269, -2.88443731}

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