4x^2+18x-31=0

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Solution for 4x^2+18x-31=0 equation:


Simplifying
4x2 + 18x + -31 = 0

Reorder the terms:
-31 + 18x + 4x2 = 0

Solving
-31 + 18x + 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 + 4.5x + x2 = 0

Move the constant term to the right:

Add '7.75' to each side of the equation.
-7.75 + 4.5x + 7.75 + x2 = 0 + 7.75

Reorder the terms:
-7.75 + 7.75 + 4.5x + x2 = 0 + 7.75

Combine like terms: -7.75 + 7.75 = 0.00
0.00 + 4.5x + x2 = 0 + 7.75
4.5x + x2 = 0 + 7.75

Combine like terms: 0 + 7.75 = 7.75
4.5x + x2 = 7.75

The x term is 4.5x.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5x + 5.0625 + x2 = 7.75 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 7.75 + 5.0625

Combine like terms: 7.75 + 5.0625 = 12.8125
5.0625 + 4.5x + x2 = 12.8125

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 12.8125

Calculate the square root of the right side: 3.579455266

Break this problem into two subproblems by setting 
(x + 2.25) equal to 3.579455266 and -3.579455266.

Subproblem 1

x + 2.25 = 3.579455266 Simplifying x + 2.25 = 3.579455266 Reorder the terms: 2.25 + x = 3.579455266 Solving 2.25 + x = 3.579455266 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 3.579455266 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 3.579455266 + -2.25 x = 3.579455266 + -2.25 Combine like terms: 3.579455266 + -2.25 = 1.329455266 x = 1.329455266 Simplifying x = 1.329455266

Subproblem 2

x + 2.25 = -3.579455266 Simplifying x + 2.25 = -3.579455266 Reorder the terms: 2.25 + x = -3.579455266 Solving 2.25 + x = -3.579455266 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -3.579455266 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -3.579455266 + -2.25 x = -3.579455266 + -2.25 Combine like terms: -3.579455266 + -2.25 = -5.829455266 x = -5.829455266 Simplifying x = -5.829455266

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.329455266, -5.829455266}

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