4x^2+24x+31=0

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Solution for 4x^2+24x+31=0 equation:


Simplifying
4x2 + 24x + 31 = 0

Reorder the terms:
31 + 24x + 4x2 = 0

Solving
31 + 24x + 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 + 6x + x2 = 0

Move the constant term to the right:

Add '-7.75' to each side of the equation.
7.75 + 6x + -7.75 + x2 = 0 + -7.75

Reorder the terms:
7.75 + -7.75 + 6x + x2 = 0 + -7.75

Combine like terms: 7.75 + -7.75 = 0.00
0.00 + 6x + x2 = 0 + -7.75
6x + x2 = 0 + -7.75

Combine like terms: 0 + -7.75 = -7.75
6x + x2 = -7.75

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = -7.75 + 9

Reorder the terms:
9 + 6x + x2 = -7.75 + 9

Combine like terms: -7.75 + 9 = 1.25
9 + 6x + x2 = 1.25

Factor a perfect square on the left side:
(x + 3)(x + 3) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + 3) equal to 1.118033989 and -1.118033989.

Subproblem 1

x + 3 = 1.118033989 Simplifying x + 3 = 1.118033989 Reorder the terms: 3 + x = 1.118033989 Solving 3 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 1.118033989 + -3 Combine like terms: 3 + -3 = 0 0 + x = 1.118033989 + -3 x = 1.118033989 + -3 Combine like terms: 1.118033989 + -3 = -1.881966011 x = -1.881966011 Simplifying x = -1.881966011

Subproblem 2

x + 3 = -1.118033989 Simplifying x + 3 = -1.118033989 Reorder the terms: 3 + x = -1.118033989 Solving 3 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -1.118033989 + -3 Combine like terms: 3 + -3 = 0 0 + x = -1.118033989 + -3 x = -1.118033989 + -3 Combine like terms: -1.118033989 + -3 = -4.118033989 x = -4.118033989 Simplifying x = -4.118033989

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.881966011, -4.118033989}

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