4x^2-12x-31=0

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Solution for 4x^2-12x-31=0 equation:


Simplifying
4x2 + -12x + -31 = 0

Reorder the terms:
-31 + -12x + 4x2 = 0

Solving
-31 + -12x + 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 + -3x + x2 = 0

Move the constant term to the right:

Add '7.75' to each side of the equation.
-7.75 + -3x + 7.75 + x2 = 0 + 7.75

Reorder the terms:
-7.75 + 7.75 + -3x + x2 = 0 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + -3x + x2 = 0 + 7.75
-3x + x2 = 0 + 7.75

Combine like terms: 0 + 7.75 = 7.75
-3x + x2 = 7.75

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 7.75 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 7.75 + 2.25

Combine like terms: 7.75 + 2.25 = 10
2.25 + -3x + x2 = 10

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(x + -1.5) equal to 3.16227766 and -3.16227766.

Subproblem 1

x + -1.5 = 3.16227766 Simplifying x + -1.5 = 3.16227766 Reorder the terms: -1.5 + x = 3.16227766 Solving -1.5 + x = 3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 3.16227766 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 3.16227766 + 1.5 x = 3.16227766 + 1.5 Combine like terms: 3.16227766 + 1.5 = 4.66227766 x = 4.66227766 Simplifying x = 4.66227766

Subproblem 2

x + -1.5 = -3.16227766 Simplifying x + -1.5 = -3.16227766 Reorder the terms: -1.5 + x = -3.16227766 Solving -1.5 + x = -3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -3.16227766 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -3.16227766 + 1.5 x = -3.16227766 + 1.5 Combine like terms: -3.16227766 + 1.5 = -1.66227766 x = -1.66227766 Simplifying x = -1.66227766

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.66227766, -1.66227766}

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