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