4x^2+7x-31=0

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Solution for 4x^2+7x-31=0 equation: 


Simplifying
4x2 + 7x + -31 = 0

Reorder the terms:
-31 + 7x + 4x2 = 0

Solving
-31 + 7x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-7.75 + 1.75x + x2 = 0

Move the constant term to the right:

Add '7.75' to each side of the equation.
-7.75 + 1.75x + 7.75 + x2 = 0 + 7.75

Reorder the terms:
-7.75 + 7.75 + 1.75x + x2 = 0 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + 1.75x + x2 = 0 + 7.75
1.75x + x2 = 0 + 7.75

Combine like terms: 0 + 7.75 = 7.75
1.75x + x2 = 7.75

The x term is 1.75x.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75x + 0.765625 + x2 = 7.75 + 0.765625

Reorder the terms:
0.765625 + 1.75x + x2 = 7.75 + 0.765625

Combine like terms: 7.75 + 0.765625 = 8.515625
0.765625 + 1.75x + x2 = 8.515625

Factor a perfect square on the left side:
(x + 0.875)(x + 0.875) = 8.515625

Calculate the square root of the right side: 2.918154382

Break this problem into two subproblems by setting 
(x + 0.875) equal to 2.918154382 and -2.918154382.


Subproblem 1
x + 0.875 = 2.918154382

Simplifying
x + 0.875 = 2.918154382

Reorder the terms:
0.875 + x = 2.918154382

Solving
0.875 + x = 2.918154382

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.875' to each side of the equation.
0.875 + -0.875 + x = 2.918154382 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + x = 2.918154382 + -0.875
x = 2.918154382 + -0.875

Combine like terms: 2.918154382 + -0.875 = 2.043154382
x = 2.043154382

Simplifying
x = 2.043154382


Subproblem 2
x + 0.875 = -2.918154382

Simplifying
x + 0.875 = -2.918154382

Reorder the terms:
0.875 + x = -2.918154382

Solving
0.875 + x = -2.918154382

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.875' to each side of the equation.
0.875 + -0.875 + x = -2.918154382 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + x = -2.918154382 + -0.875
x = -2.918154382 + -0.875

Combine like terms: -2.918154382 + -0.875 = -3.793154382
x = -3.793154382

Simplifying
x = -3.793154382


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.043154382, -3.793154382}

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