2x^2-13x-31=0

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Solution for 2x^2-13x-31=0 equation:


Simplifying
2x2 + -13x + -31 = 0

Reorder the terms:
-31 + -13x + 2x2 = 0

Solving
-31 + -13x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-15.5 + -6.5x + x2 = 0

Move the constant term to the right:

Add '15.5' to each side of the equation.
-15.5 + -6.5x + 15.5 + x2 = 0 + 15.5

Reorder the terms:
-15.5 + 15.5 + -6.5x + x2 = 0 + 15.5

Combine like terms: -15.5 + 15.5 = 0.0
0.0 + -6.5x + x2 = 0 + 15.5
-6.5x + x2 = 0 + 15.5

Combine like terms: 0 + 15.5 = 15.5
-6.5x + x2 = 15.5

The x term is -6.5x.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5x + 10.5625 + x2 = 15.5 + 10.5625

Reorder the terms:
10.5625 + -6.5x + x2 = 15.5 + 10.5625

Combine like terms: 15.5 + 10.5625 = 26.0625
10.5625 + -6.5x + x2 = 26.0625

Factor a perfect square on the left side:
(x + -3.25)(x + -3.25) = 26.0625

Calculate the square root of the right side: 5.105144464

Break this problem into two subproblems by setting 
(x + -3.25) equal to 5.105144464 and -5.105144464.

Subproblem 1

x + -3.25 = 5.105144464 Simplifying x + -3.25 = 5.105144464 Reorder the terms: -3.25 + x = 5.105144464 Solving -3.25 + x = 5.105144464 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = 5.105144464 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = 5.105144464 + 3.25 x = 5.105144464 + 3.25 Combine like terms: 5.105144464 + 3.25 = 8.355144464 x = 8.355144464 Simplifying x = 8.355144464

Subproblem 2

x + -3.25 = -5.105144464 Simplifying x + -3.25 = -5.105144464 Reorder the terms: -3.25 + x = -5.105144464 Solving -3.25 + x = -5.105144464 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + x = -5.105144464 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + x = -5.105144464 + 3.25 x = -5.105144464 + 3.25 Combine like terms: -5.105144464 + 3.25 = -1.855144464 x = -1.855144464 Simplifying x = -1.855144464

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.355144464, -1.855144464}

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