3x^2+15x+13=0

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Solution for 3x^2+15x+13=0 equation:


Simplifying
3x2 + 15x + 13 = 0

Reorder the terms:
13 + 15x + 3x2 = 0

Solving
13 + 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'.
4.333333333 + 5x + x2 = 0

Move the constant term to the right:

Add '-4.333333333' to each side of the equation.
4.333333333 + 5x + -4.333333333 + x2 = 0 + -4.333333333

Reorder the terms:
4.333333333 + -4.333333333 + 5x + x2 = 0 + -4.333333333

Combine like terms: 4.333333333 + -4.333333333 = 0.000000000
0.000000000 + 5x + x2 = 0 + -4.333333333
5x + x2 = 0 + -4.333333333

Combine like terms: 0 + -4.333333333 = -4.333333333
5x + x2 = -4.333333333

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 = -4.333333333 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -4.333333333 + 6.25

Combine like terms: -4.333333333 + 6.25 = 1.916666667
6.25 + 5x + x2 = 1.916666667

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 1.916666667

Calculate the square root of the right side: 1.384437311

Break this problem into two subproblems by setting 
(x + 2.5) equal to 1.384437311 and -1.384437311.

Subproblem 1

x + 2.5 = 1.384437311 Simplifying x + 2.5 = 1.384437311 Reorder the terms: 2.5 + x = 1.384437311 Solving 2.5 + x = 1.384437311 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 = 1.384437311 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 1.384437311 + -2.5 x = 1.384437311 + -2.5 Combine like terms: 1.384437311 + -2.5 = -1.115562689 x = -1.115562689 Simplifying x = -1.115562689

Subproblem 2

x + 2.5 = -1.384437311 Simplifying x + 2.5 = -1.384437311 Reorder the terms: 2.5 + x = -1.384437311 Solving 2.5 + x = -1.384437311 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 = -1.384437311 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -1.384437311 + -2.5 x = -1.384437311 + -2.5 Combine like terms: -1.384437311 + -2.5 = -3.884437311 x = -3.884437311 Simplifying x = -3.884437311

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.115562689, -3.884437311}

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