2x^2+16x+31=0

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Solution for 2x^2+16x+31=0 equation:


Simplifying
2x2 + 16x + 31 = 0

Reorder the terms:
31 + 16x + 2x2 = 0

Solving
31 + 16x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
15.5 + 8x + x2 = 0

Move the constant term to the right:

Add '-15.5' to each side of the equation.
15.5 + 8x + -15.5 + x2 = 0 + -15.5

Reorder the terms:
15.5 + -15.5 + 8x + x2 = 0 + -15.5

Combine like terms: 15.5 + -15.5 = 0.0
0.0 + 8x + x2 = 0 + -15.5
8x + x2 = 0 + -15.5

Combine like terms: 0 + -15.5 = -15.5
8x + x2 = -15.5

The x term is 8x.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8x + 16 + x2 = -15.5 + 16

Reorder the terms:
16 + 8x + x2 = -15.5 + 16

Combine like terms: -15.5 + 16 = 0.5
16 + 8x + x2 = 0.5

Factor a perfect square on the left side:
(x + 4)(x + 4) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + 4) equal to 0.707106781 and -0.707106781.

Subproblem 1

x + 4 = 0.707106781 Simplifying x + 4 = 0.707106781 Reorder the terms: 4 + x = 0.707106781 Solving 4 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = 0.707106781 + -4 Combine like terms: 4 + -4 = 0 0 + x = 0.707106781 + -4 x = 0.707106781 + -4 Combine like terms: 0.707106781 + -4 = -3.292893219 x = -3.292893219 Simplifying x = -3.292893219

Subproblem 2

x + 4 = -0.707106781 Simplifying x + 4 = -0.707106781 Reorder the terms: 4 + x = -0.707106781 Solving 4 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = -0.707106781 + -4 Combine like terms: 4 + -4 = 0 0 + x = -0.707106781 + -4 x = -0.707106781 + -4 Combine like terms: -0.707106781 + -4 = -4.707106781 x = -4.707106781 Simplifying x = -4.707106781

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.292893219, -4.707106781}

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