2x^2-x-31=31

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Solution for 2x^2-x-31=31 equation:


Simplifying
2x2 + -1x + -31 = 31

Reorder the terms:
-31 + -1x + 2x2 = 31

Solving
-31 + -1x + 2x2 = 31

Solving for variable 'x'.

Reorder the terms:
-31 + -31 + -1x + 2x2 = 31 + -31

Combine like terms: -31 + -31 = -62
-62 + -1x + 2x2 = 31 + -31

Combine like terms: 31 + -31 = 0
-62 + -1x + 2x2 = 0

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

Divide each side by '2'.
-31 + -0.5x + x2 = 0

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + -0.5x + 31 + x2 = 0 + 31

Reorder the terms:
-31 + 31 + -0.5x + x2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + -0.5x + x2 = 0 + 31
-0.5x + x2 = 0 + 31

Combine like terms: 0 + 31 = 31
-0.5x + x2 = 31

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 31 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 31 + 0.0625

Combine like terms: 31 + 0.0625 = 31.0625
0.0625 + -0.5x + x2 = 31.0625

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 31.0625

Calculate the square root of the right side: 5.573374202

Break this problem into two subproblems by setting 
(x + -0.25) equal to 5.573374202 and -5.573374202.

Subproblem 1

x + -0.25 = 5.573374202 Simplifying x + -0.25 = 5.573374202 Reorder the terms: -0.25 + x = 5.573374202 Solving -0.25 + x = 5.573374202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 5.573374202 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 5.573374202 + 0.25 x = 5.573374202 + 0.25 Combine like terms: 5.573374202 + 0.25 = 5.823374202 x = 5.823374202 Simplifying x = 5.823374202

Subproblem 2

x + -0.25 = -5.573374202 Simplifying x + -0.25 = -5.573374202 Reorder the terms: -0.25 + x = -5.573374202 Solving -0.25 + x = -5.573374202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -5.573374202 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -5.573374202 + 0.25 x = -5.573374202 + 0.25 Combine like terms: -5.573374202 + 0.25 = -5.323374202 x = -5.323374202 Simplifying x = -5.323374202

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.823374202, -5.323374202}

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