3x^2+15x+11=0

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Solution for 3x^2+15x+11=0 equation:


Simplifying
3x2 + 15x + 11 = 0

Reorder the terms:
11 + 15x + 3x2 = 0

Solving
11 + 15x + 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'.
3.666666667 + 5x + x2 = 0

Move the constant term to the right:

Add '-3.666666667' to each side of the equation.
3.666666667 + 5x + -3.666666667 + x2 = 0 + -3.666666667

Reorder the terms:
3.666666667 + -3.666666667 + 5x + x2 = 0 + -3.666666667

Combine like terms: 3.666666667 + -3.666666667 = 0.000000000
0.000000000 + 5x + x2 = 0 + -3.666666667
5x + x2 = 0 + -3.666666667

Combine like terms: 0 + -3.666666667 = -3.666666667
5x + x2 = -3.666666667

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = -3.666666667 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -3.666666667 + 6.25

Combine like terms: -3.666666667 + 6.25 = 2.583333333
6.25 + 5x + x2 = 2.583333333

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 2.583333333

Calculate the square root of the right side: 1.607275127

Break this problem into two subproblems by setting 
(x + 2.5) equal to 1.607275127 and -1.607275127.

Subproblem 1

x + 2.5 = 1.607275127 Simplifying x + 2.5 = 1.607275127 Reorder the terms: 2.5 + x = 1.607275127 Solving 2.5 + x = 1.607275127 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 1.607275127 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 1.607275127 + -2.5 x = 1.607275127 + -2.5 Combine like terms: 1.607275127 + -2.5 = -0.892724873 x = -0.892724873 Simplifying x = -0.892724873

Subproblem 2

x + 2.5 = -1.607275127 Simplifying x + 2.5 = -1.607275127 Reorder the terms: 2.5 + x = -1.607275127 Solving 2.5 + x = -1.607275127 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -1.607275127 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -1.607275127 + -2.5 x = -1.607275127 + -2.5 Combine like terms: -1.607275127 + -2.5 = -4.107275127 x = -4.107275127 Simplifying x = -4.107275127

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.892724873, -4.107275127}

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