4x^2+31x+1=0

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Solution for 4x^2+31x+1=0 equation:


Simplifying
4x2 + 31x + 1 = 0

Reorder the terms:
1 + 31x + 4x2 = 0

Solving
1 + 31x + 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'.
0.25 + 7.75x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 7.75x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 7.75x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 7.75x + x2 = 0 + -0.25
7.75x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
7.75x + x2 = -0.25

The x term is 7.75x.  Take half its coefficient (3.875).
Square it (15.015625) and add it to both sides.

Add '15.015625' to each side of the equation.
7.75x + 15.015625 + x2 = -0.25 + 15.015625

Reorder the terms:
15.015625 + 7.75x + x2 = -0.25 + 15.015625

Combine like terms: -0.25 + 15.015625 = 14.765625
15.015625 + 7.75x + x2 = 14.765625

Factor a perfect square on the left side:
(x + 3.875)(x + 3.875) = 14.765625

Calculate the square root of the right side: 3.842606537

Break this problem into two subproblems by setting 
(x + 3.875) equal to 3.842606537 and -3.842606537.

Subproblem 1

x + 3.875 = 3.842606537 Simplifying x + 3.875 = 3.842606537 Reorder the terms: 3.875 + x = 3.842606537 Solving 3.875 + x = 3.842606537 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = 3.842606537 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = 3.842606537 + -3.875 x = 3.842606537 + -3.875 Combine like terms: 3.842606537 + -3.875 = -0.032393463 x = -0.032393463 Simplifying x = -0.032393463

Subproblem 2

x + 3.875 = -3.842606537 Simplifying x + 3.875 = -3.842606537 Reorder the terms: 3.875 + x = -3.842606537 Solving 3.875 + x = -3.842606537 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = -3.842606537 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = -3.842606537 + -3.875 x = -3.842606537 + -3.875 Combine like terms: -3.842606537 + -3.875 = -7.717606537 x = -7.717606537 Simplifying x = -7.717606537

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.032393463, -7.717606537}

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