15x^2+31x+1=0

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Solution for 15x^2+31x+1=0 equation:


Simplifying
15x2 + 31x + 1 = 0

Reorder the terms:
1 + 31x + 15x2 = 0

Solving
1 + 31x + 15x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
15 the coefficient of the squared term: 

Divide each side by '15'.
0.06666666667 + 2.066666667x + x2 = 0

Move the constant term to the right:

Add '-0.06666666667' to each side of the equation.
0.06666666667 + 2.066666667x + -0.06666666667 + x2 = 0 + -0.06666666667

Reorder the terms:
0.06666666667 + -0.06666666667 + 2.066666667x + x2 = 0 + -0.06666666667

Combine like terms: 0.06666666667 + -0.06666666667 = 0.00000000000
0.00000000000 + 2.066666667x + x2 = 0 + -0.06666666667
2.066666667x + x2 = 0 + -0.06666666667

Combine like terms: 0 + -0.06666666667 = -0.06666666667
2.066666667x + x2 = -0.06666666667

The x term is 2.066666667x.  Take half its coefficient (1.033333334).
Square it (1.067777779) and add it to both sides.

Add '1.067777779' to each side of the equation.
2.066666667x + 1.067777779 + x2 = -0.06666666667 + 1.067777779

Reorder the terms:
1.067777779 + 2.066666667x + x2 = -0.06666666667 + 1.067777779

Combine like terms: -0.06666666667 + 1.067777779 = 1.00111111233
1.067777779 + 2.066666667x + x2 = 1.00111111233

Factor a perfect square on the left side:
(x + 1.033333334)(x + 1.033333334) = 1.00111111233

Calculate the square root of the right side: 1.000555402

Break this problem into two subproblems by setting 
(x + 1.033333334) equal to 1.000555402 and -1.000555402.

Subproblem 1

x + 1.033333334 = 1.000555402 Simplifying x + 1.033333334 = 1.000555402 Reorder the terms: 1.033333334 + x = 1.000555402 Solving 1.033333334 + x = 1.000555402 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + x = 1.000555402 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + x = 1.000555402 + -1.033333334 x = 1.000555402 + -1.033333334 Combine like terms: 1.000555402 + -1.033333334 = -0.032777932 x = -0.032777932 Simplifying x = -0.032777932

Subproblem 2

x + 1.033333334 = -1.000555402 Simplifying x + 1.033333334 = -1.000555402 Reorder the terms: 1.033333334 + x = -1.000555402 Solving 1.033333334 + x = -1.000555402 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.033333334' to each side of the equation. 1.033333334 + -1.033333334 + x = -1.000555402 + -1.033333334 Combine like terms: 1.033333334 + -1.033333334 = 0.000000000 0.000000000 + x = -1.000555402 + -1.033333334 x = -1.000555402 + -1.033333334 Combine like terms: -1.000555402 + -1.033333334 = -2.033888736 x = -2.033888736 Simplifying x = -2.033888736

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.032777932, -2.033888736}

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