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Simplifying -2x2 + -17x + -31 = 0 Reorder the terms: -31 + -17x + -2x2 = 0 Solving -31 + -17x + -2x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '-1'. -1(31 + 17x + 2x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(31 + 17x + 2x2)' equal to zero and attempt to solve: Simplifying 31 + 17x + 2x2 = 0 Solving 31 + 17x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 15.5 + 8.5x + x2 = 0 Move the constant term to the right: Add '-15.5' to each side of the equation. 15.5 + 8.5x + -15.5 + x2 = 0 + -15.5 Reorder the terms: 15.5 + -15.5 + 8.5x + x2 = 0 + -15.5 Combine like terms: 15.5 + -15.5 = 0.0 0.0 + 8.5x + x2 = 0 + -15.5 8.5x + x2 = 0 + -15.5 Combine like terms: 0 + -15.5 = -15.5 8.5x + x2 = -15.5 The x term is 8.5x. Take half its coefficient (4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. 8.5x + 18.0625 + x2 = -15.5 + 18.0625 Reorder the terms: 18.0625 + 8.5x + x2 = -15.5 + 18.0625 Combine like terms: -15.5 + 18.0625 = 2.5625 18.0625 + 8.5x + x2 = 2.5625 Factor a perfect square on the left side: (x + 4.25)(x + 4.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (x + 4.25) equal to 1.600781059 and -1.600781059.Subproblem 1
x + 4.25 = 1.600781059 Simplifying x + 4.25 = 1.600781059 Reorder the terms: 4.25 + x = 1.600781059 Solving 4.25 + x = 1.600781059 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = 1.600781059 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = 1.600781059 + -4.25 x = 1.600781059 + -4.25 Combine like terms: 1.600781059 + -4.25 = -2.649218941 x = -2.649218941 Simplifying x = -2.649218941Subproblem 2
x + 4.25 = -1.600781059 Simplifying x + 4.25 = -1.600781059 Reorder the terms: 4.25 + x = -1.600781059 Solving 4.25 + x = -1.600781059 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = -1.600781059 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = -1.600781059 + -4.25 x = -1.600781059 + -4.25 Combine like terms: -1.600781059 + -4.25 = -5.850781059 x = -5.850781059 Simplifying x = -5.850781059Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2.649218941, -5.850781059}Solution
x = {-2.649218941, -5.850781059}
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