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Simplifying 9x2 + 31x + -108 = 0 Reorder the terms: -108 + 31x + 9x2 = 0 Solving -108 + 31x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -12 + 3.444444444x + x2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 3.444444444x + 12 + x2 = 0 + 12 Reorder the terms: -12 + 12 + 3.444444444x + x2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 3.444444444x + x2 = 0 + 12 3.444444444x + x2 = 0 + 12 Combine like terms: 0 + 12 = 12 3.444444444x + x2 = 12 The x term is 3.444444444x. Take half its coefficient (1.722222222). Square it (2.966049382) and add it to both sides. Add '2.966049382' to each side of the equation. 3.444444444x + 2.966049382 + x2 = 12 + 2.966049382 Reorder the terms: 2.966049382 + 3.444444444x + x2 = 12 + 2.966049382 Combine like terms: 12 + 2.966049382 = 14.966049382 2.966049382 + 3.444444444x + x2 = 14.966049382 Factor a perfect square on the left side: (x + 1.722222222)(x + 1.722222222) = 14.966049382 Calculate the square root of the right side: 3.868597857 Break this problem into two subproblems by setting (x + 1.722222222) equal to 3.868597857 and -3.868597857.Subproblem 1
x + 1.722222222 = 3.868597857 Simplifying x + 1.722222222 = 3.868597857 Reorder the terms: 1.722222222 + x = 3.868597857 Solving 1.722222222 + x = 3.868597857 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.722222222' to each side of the equation. 1.722222222 + -1.722222222 + x = 3.868597857 + -1.722222222 Combine like terms: 1.722222222 + -1.722222222 = 0.000000000 0.000000000 + x = 3.868597857 + -1.722222222 x = 3.868597857 + -1.722222222 Combine like terms: 3.868597857 + -1.722222222 = 2.146375635 x = 2.146375635 Simplifying x = 2.146375635Subproblem 2
x + 1.722222222 = -3.868597857 Simplifying x + 1.722222222 = -3.868597857 Reorder the terms: 1.722222222 + x = -3.868597857 Solving 1.722222222 + x = -3.868597857 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.722222222' to each side of the equation. 1.722222222 + -1.722222222 + x = -3.868597857 + -1.722222222 Combine like terms: 1.722222222 + -1.722222222 = 0.000000000 0.000000000 + x = -3.868597857 + -1.722222222 x = -3.868597857 + -1.722222222 Combine like terms: -3.868597857 + -1.722222222 = -5.590820079 x = -5.590820079 Simplifying x = -5.590820079Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.146375635, -5.590820079}
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