7x^2+31x+5=0

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Solution for 7x^2+31x+5=0 equation:


Simplifying
7x2 + 31x + 5 = 0

Reorder the terms:
5 + 31x + 7x2 = 0

Solving
5 + 31x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
0.7142857143 + 4.428571429x + x2 = 0

Move the constant term to the right:

Add '-0.7142857143' to each side of the equation.
0.7142857143 + 4.428571429x + -0.7142857143 + x2 = 0 + -0.7142857143

Reorder the terms:
0.7142857143 + -0.7142857143 + 4.428571429x + x2 = 0 + -0.7142857143

Combine like terms: 0.7142857143 + -0.7142857143 = 0.0000000000
0.0000000000 + 4.428571429x + x2 = 0 + -0.7142857143
4.428571429x + x2 = 0 + -0.7142857143

Combine like terms: 0 + -0.7142857143 = -0.7142857143
4.428571429x + x2 = -0.7142857143

The x term is 4.428571429x.  Take half its coefficient (2.214285715).
Square it (4.903061228) and add it to both sides.

Add '4.903061228' to each side of the equation.
4.428571429x + 4.903061228 + x2 = -0.7142857143 + 4.903061228

Reorder the terms:
4.903061228 + 4.428571429x + x2 = -0.7142857143 + 4.903061228

Combine like terms: -0.7142857143 + 4.903061228 = 4.1887755137
4.903061228 + 4.428571429x + x2 = 4.1887755137

Factor a perfect square on the left side:
(x + 2.214285715)(x + 2.214285715) = 4.1887755137

Calculate the square root of the right side: 2.046649827

Break this problem into two subproblems by setting 
(x + 2.214285715) equal to 2.046649827 and -2.046649827.

Subproblem 1

x + 2.214285715 = 2.046649827 Simplifying x + 2.214285715 = 2.046649827 Reorder the terms: 2.214285715 + x = 2.046649827 Solving 2.214285715 + x = 2.046649827 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.214285715' to each side of the equation. 2.214285715 + -2.214285715 + x = 2.046649827 + -2.214285715 Combine like terms: 2.214285715 + -2.214285715 = 0.000000000 0.000000000 + x = 2.046649827 + -2.214285715 x = 2.046649827 + -2.214285715 Combine like terms: 2.046649827 + -2.214285715 = -0.167635888 x = -0.167635888 Simplifying x = -0.167635888

Subproblem 2

x + 2.214285715 = -2.046649827 Simplifying x + 2.214285715 = -2.046649827 Reorder the terms: 2.214285715 + x = -2.046649827 Solving 2.214285715 + x = -2.046649827 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.214285715' to each side of the equation. 2.214285715 + -2.214285715 + x = -2.046649827 + -2.214285715 Combine like terms: 2.214285715 + -2.214285715 = 0.000000000 0.000000000 + x = -2.046649827 + -2.214285715 x = -2.046649827 + -2.214285715 Combine like terms: -2.046649827 + -2.214285715 = -4.260935542 x = -4.260935542 Simplifying x = -4.260935542

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.167635888, -4.260935542}

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