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Simplifying 9x2 + 29x + -28 = 1 Reorder the terms: -28 + 29x + 9x2 = 1 Solving -28 + 29x + 9x2 = 1 Solving for variable 'x'. Reorder the terms: -28 + -1 + 29x + 9x2 = 1 + -1 Combine like terms: -28 + -1 = -29 -29 + 29x + 9x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -29 + 29x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -3.222222222 + 3.222222222x + x2 = 0 Move the constant term to the right: Add '3.222222222' to each side of the equation. -3.222222222 + 3.222222222x + 3.222222222 + x2 = 0 + 3.222222222 Reorder the terms: -3.222222222 + 3.222222222 + 3.222222222x + x2 = 0 + 3.222222222 Combine like terms: -3.222222222 + 3.222222222 = 0.000000000 0.000000000 + 3.222222222x + x2 = 0 + 3.222222222 3.222222222x + x2 = 0 + 3.222222222 Combine like terms: 0 + 3.222222222 = 3.222222222 3.222222222x + x2 = 3.222222222 The x term is 3.222222222x. Take half its coefficient (1.611111111). Square it (2.595679012) and add it to both sides. Add '2.595679012' to each side of the equation. 3.222222222x + 2.595679012 + x2 = 3.222222222 + 2.595679012 Reorder the terms: 2.595679012 + 3.222222222x + x2 = 3.222222222 + 2.595679012 Combine like terms: 3.222222222 + 2.595679012 = 5.817901234 2.595679012 + 3.222222222x + x2 = 5.817901234 Factor a perfect square on the left side: (x + 1.611111111)(x + 1.611111111) = 5.817901234 Calculate the square root of the right side: 2.412032594 Break this problem into two subproblems by setting (x + 1.611111111) equal to 2.412032594 and -2.412032594.Subproblem 1
x + 1.611111111 = 2.412032594 Simplifying x + 1.611111111 = 2.412032594 Reorder the terms: 1.611111111 + x = 2.412032594 Solving 1.611111111 + x = 2.412032594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.611111111' to each side of the equation. 1.611111111 + -1.611111111 + x = 2.412032594 + -1.611111111 Combine like terms: 1.611111111 + -1.611111111 = 0.000000000 0.000000000 + x = 2.412032594 + -1.611111111 x = 2.412032594 + -1.611111111 Combine like terms: 2.412032594 + -1.611111111 = 0.800921483 x = 0.800921483 Simplifying x = 0.800921483Subproblem 2
x + 1.611111111 = -2.412032594 Simplifying x + 1.611111111 = -2.412032594 Reorder the terms: 1.611111111 + x = -2.412032594 Solving 1.611111111 + x = -2.412032594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.611111111' to each side of the equation. 1.611111111 + -1.611111111 + x = -2.412032594 + -1.611111111 Combine like terms: 1.611111111 + -1.611111111 = 0.000000000 0.000000000 + x = -2.412032594 + -1.611111111 x = -2.412032594 + -1.611111111 Combine like terms: -2.412032594 + -1.611111111 = -4.023143705 x = -4.023143705 Simplifying x = -4.023143705Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.800921483, -4.023143705}
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