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Simplifying 106x2 + 743x + -23478 = 0 Reorder the terms: -23478 + 743x + 106x2 = 0 Solving -23478 + 743x + 106x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 106 the coefficient of the squared term: Divide each side by '106'. -221.490566 + 7.009433962x + x2 = 0 Move the constant term to the right: Add '221.490566' to each side of the equation. -221.490566 + 7.009433962x + 221.490566 + x2 = 0 + 221.490566 Reorder the terms: -221.490566 + 221.490566 + 7.009433962x + x2 = 0 + 221.490566 Combine like terms: -221.490566 + 221.490566 = 0.000000 0.000000 + 7.009433962x + x2 = 0 + 221.490566 7.009433962x + x2 = 0 + 221.490566 Combine like terms: 0 + 221.490566 = 221.490566 7.009433962x + x2 = 221.490566 The x term is 7.009433962x. Take half its coefficient (3.504716981). Square it (12.28304112) and add it to both sides. Add '12.28304112' to each side of the equation. 7.009433962x + 12.28304112 + x2 = 221.490566 + 12.28304112 Reorder the terms: 12.28304112 + 7.009433962x + x2 = 221.490566 + 12.28304112 Combine like terms: 221.490566 + 12.28304112 = 233.77360712 12.28304112 + 7.009433962x + x2 = 233.77360712 Factor a perfect square on the left side: (x + 3.504716981)(x + 3.504716981) = 233.77360712 Calculate the square root of the right side: 15.289656867 Break this problem into two subproblems by setting (x + 3.504716981) equal to 15.289656867 and -15.289656867.Subproblem 1
x + 3.504716981 = 15.289656867 Simplifying x + 3.504716981 = 15.289656867 Reorder the terms: 3.504716981 + x = 15.289656867 Solving 3.504716981 + x = 15.289656867 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.504716981' to each side of the equation. 3.504716981 + -3.504716981 + x = 15.289656867 + -3.504716981 Combine like terms: 3.504716981 + -3.504716981 = 0.000000000 0.000000000 + x = 15.289656867 + -3.504716981 x = 15.289656867 + -3.504716981 Combine like terms: 15.289656867 + -3.504716981 = 11.784939886 x = 11.784939886 Simplifying x = 11.784939886Subproblem 2
x + 3.504716981 = -15.289656867 Simplifying x + 3.504716981 = -15.289656867 Reorder the terms: 3.504716981 + x = -15.289656867 Solving 3.504716981 + x = -15.289656867 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.504716981' to each side of the equation. 3.504716981 + -3.504716981 + x = -15.289656867 + -3.504716981 Combine like terms: 3.504716981 + -3.504716981 = 0.000000000 0.000000000 + x = -15.289656867 + -3.504716981 x = -15.289656867 + -3.504716981 Combine like terms: -15.289656867 + -3.504716981 = -18.794373848 x = -18.794373848 Simplifying x = -18.794373848Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.784939886, -18.794373848}
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