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Simplifying 9x2 + -3x + -23 = 0 Reorder the terms: -23 + -3x + 9x2 = 0 Solving -23 + -3x + 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'. -2.555555556 + -0.3333333333x + x2 = 0 Move the constant term to the right: Add '2.555555556' to each side of the equation. -2.555555556 + -0.3333333333x + 2.555555556 + x2 = 0 + 2.555555556 Reorder the terms: -2.555555556 + 2.555555556 + -0.3333333333x + x2 = 0 + 2.555555556 Combine like terms: -2.555555556 + 2.555555556 = 0.000000000 0.000000000 + -0.3333333333x + x2 = 0 + 2.555555556 -0.3333333333x + x2 = 0 + 2.555555556 Combine like terms: 0 + 2.555555556 = 2.555555556 -0.3333333333x + x2 = 2.555555556 The x term is -0.3333333333x. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333x + 0.02777777779 + x2 = 2.555555556 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333x + x2 = 2.555555556 + 0.02777777779 Combine like terms: 2.555555556 + 0.02777777779 = 2.58333333379 0.02777777779 + -0.3333333333x + x2 = 2.58333333379 Factor a perfect square on the left side: (x + -0.1666666667)(x + -0.1666666667) = 2.58333333379 Calculate the square root of the right side: 1.607275127 Break this problem into two subproblems by setting (x + -0.1666666667) equal to 1.607275127 and -1.607275127.Subproblem 1
x + -0.1666666667 = 1.607275127 Simplifying x + -0.1666666667 = 1.607275127 Reorder the terms: -0.1666666667 + x = 1.607275127 Solving -0.1666666667 + x = 1.607275127 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + x = 1.607275127 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = 1.607275127 + 0.1666666667 x = 1.607275127 + 0.1666666667 Combine like terms: 1.607275127 + 0.1666666667 = 1.7739417937 x = 1.7739417937 Simplifying x = 1.7739417937Subproblem 2
x + -0.1666666667 = -1.607275127 Simplifying x + -0.1666666667 = -1.607275127 Reorder the terms: -0.1666666667 + x = -1.607275127 Solving -0.1666666667 + x = -1.607275127 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + x = -1.607275127 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + x = -1.607275127 + 0.1666666667 x = -1.607275127 + 0.1666666667 Combine like terms: -1.607275127 + 0.1666666667 = -1.4406084603 x = -1.4406084603 Simplifying x = -1.4406084603Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.7739417937, -1.4406084603}
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