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Simplifying 31x2 + -118x + -180 = 0 Reorder the terms: -180 + -118x + 31x2 = 0 Solving -180 + -118x + 31x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 31 the coefficient of the squared term: Divide each side by '31'. -5.806451613 + -3.806451613x + x2 = 0 Move the constant term to the right: Add '5.806451613' to each side of the equation. -5.806451613 + -3.806451613x + 5.806451613 + x2 = 0 + 5.806451613 Reorder the terms: -5.806451613 + 5.806451613 + -3.806451613x + x2 = 0 + 5.806451613 Combine like terms: -5.806451613 + 5.806451613 = 0.000000000 0.000000000 + -3.806451613x + x2 = 0 + 5.806451613 -3.806451613x + x2 = 0 + 5.806451613 Combine like terms: 0 + 5.806451613 = 5.806451613 -3.806451613x + x2 = 5.806451613 The x term is -3.806451613x. Take half its coefficient (-1.903225807). Square it (3.622268472) and add it to both sides. Add '3.622268472' to each side of the equation. -3.806451613x + 3.622268472 + x2 = 5.806451613 + 3.622268472 Reorder the terms: 3.622268472 + -3.806451613x + x2 = 5.806451613 + 3.622268472 Combine like terms: 5.806451613 + 3.622268472 = 9.428720085 3.622268472 + -3.806451613x + x2 = 9.428720085 Factor a perfect square on the left side: (x + -1.903225807)(x + -1.903225807) = 9.428720085 Calculate the square root of the right side: 3.070622101 Break this problem into two subproblems by setting (x + -1.903225807) equal to 3.070622101 and -3.070622101.Subproblem 1
x + -1.903225807 = 3.070622101 Simplifying x + -1.903225807 = 3.070622101 Reorder the terms: -1.903225807 + x = 3.070622101 Solving -1.903225807 + x = 3.070622101 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.903225807' to each side of the equation. -1.903225807 + 1.903225807 + x = 3.070622101 + 1.903225807 Combine like terms: -1.903225807 + 1.903225807 = 0.000000000 0.000000000 + x = 3.070622101 + 1.903225807 x = 3.070622101 + 1.903225807 Combine like terms: 3.070622101 + 1.903225807 = 4.973847908 x = 4.973847908 Simplifying x = 4.973847908Subproblem 2
x + -1.903225807 = -3.070622101 Simplifying x + -1.903225807 = -3.070622101 Reorder the terms: -1.903225807 + x = -3.070622101 Solving -1.903225807 + x = -3.070622101 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.903225807' to each side of the equation. -1.903225807 + 1.903225807 + x = -3.070622101 + 1.903225807 Combine like terms: -1.903225807 + 1.903225807 = 0.000000000 0.000000000 + x = -3.070622101 + 1.903225807 x = -3.070622101 + 1.903225807 Combine like terms: -3.070622101 + 1.903225807 = -1.167396294 x = -1.167396294 Simplifying x = -1.167396294Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.973847908, -1.167396294}
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