8x^2+68x+31=0

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Solution for 8x^2+68x+31=0 equation:


Simplifying
8x2 + 68x + 31 = 0

Reorder the terms:
31 + 68x + 8x2 = 0

Solving
31 + 68x + 8x2 = 0

Solving for variable 'x'.

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

Divide each side by '8'.
3.875 + 8.5x + x2 = 0

Move the constant term to the right:

Add '-3.875' to each side of the equation.
3.875 + 8.5x + -3.875 + x2 = 0 + -3.875

Reorder the terms:
3.875 + -3.875 + 8.5x + x2 = 0 + -3.875

Combine like terms: 3.875 + -3.875 = 0.000
0.000 + 8.5x + x2 = 0 + -3.875
8.5x + x2 = 0 + -3.875

Combine like terms: 0 + -3.875 = -3.875
8.5x + x2 = -3.875

The x term is 8.5x.  Take half its coefficient (4.25).
Square it (18.0625) and add it to both sides.

Add '18.0625' to each side of the equation.
8.5x + 18.0625 + x2 = -3.875 + 18.0625

Reorder the terms:
18.0625 + 8.5x + x2 = -3.875 + 18.0625

Combine like terms: -3.875 + 18.0625 = 14.1875
18.0625 + 8.5x + x2 = 14.1875

Factor a perfect square on the left side:
(x + 4.25)(x + 4.25) = 14.1875

Calculate the square root of the right side: 3.766629793

Break this problem into two subproblems by setting 
(x + 4.25) equal to 3.766629793 and -3.766629793.

Subproblem 1

x + 4.25 = 3.766629793 Simplifying x + 4.25 = 3.766629793 Reorder the terms: 4.25 + x = 3.766629793 Solving 4.25 + x = 3.766629793 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = 3.766629793 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = 3.766629793 + -4.25 x = 3.766629793 + -4.25 Combine like terms: 3.766629793 + -4.25 = -0.483370207 x = -0.483370207 Simplifying x = -0.483370207

Subproblem 2

x + 4.25 = -3.766629793 Simplifying x + 4.25 = -3.766629793 Reorder the terms: 4.25 + x = -3.766629793 Solving 4.25 + x = -3.766629793 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = -3.766629793 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = -3.766629793 + -4.25 x = -3.766629793 + -4.25 Combine like terms: -3.766629793 + -4.25 = -8.016629793 x = -8.016629793 Simplifying x = -8.016629793

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.483370207, -8.016629793}

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