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Simplifying 6x2 + 13x + -22 = 102 Reorder the terms: -22 + 13x + 6x2 = 102 Solving -22 + 13x + 6x2 = 102 Solving for variable 'x'. Reorder the terms: -22 + -102 + 13x + 6x2 = 102 + -102 Combine like terms: -22 + -102 = -124 -124 + 13x + 6x2 = 102 + -102 Combine like terms: 102 + -102 = 0 -124 + 13x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -20.66666667 + 2.166666667x + x2 = 0 Move the constant term to the right: Add '20.66666667' to each side of the equation. -20.66666667 + 2.166666667x + 20.66666667 + x2 = 0 + 20.66666667 Reorder the terms: -20.66666667 + 20.66666667 + 2.166666667x + x2 = 0 + 20.66666667 Combine like terms: -20.66666667 + 20.66666667 = 0.00000000 0.00000000 + 2.166666667x + x2 = 0 + 20.66666667 2.166666667x + x2 = 0 + 20.66666667 Combine like terms: 0 + 20.66666667 = 20.66666667 2.166666667x + x2 = 20.66666667 The x term is 2.166666667x. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667x + 1.173611113 + x2 = 20.66666667 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667x + x2 = 20.66666667 + 1.173611113 Combine like terms: 20.66666667 + 1.173611113 = 21.840277783 1.173611113 + 2.166666667x + x2 = 21.840277783 Factor a perfect square on the left side: (x + 1.083333334)(x + 1.083333334) = 21.840277783 Calculate the square root of the right side: 4.673358298 Break this problem into two subproblems by setting (x + 1.083333334) equal to 4.673358298 and -4.673358298.Subproblem 1
x + 1.083333334 = 4.673358298 Simplifying x + 1.083333334 = 4.673358298 Reorder the terms: 1.083333334 + x = 4.673358298 Solving 1.083333334 + x = 4.673358298 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + x = 4.673358298 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + x = 4.673358298 + -1.083333334 x = 4.673358298 + -1.083333334 Combine like terms: 4.673358298 + -1.083333334 = 3.590024964 x = 3.590024964 Simplifying x = 3.590024964Subproblem 2
x + 1.083333334 = -4.673358298 Simplifying x + 1.083333334 = -4.673358298 Reorder the terms: 1.083333334 + x = -4.673358298 Solving 1.083333334 + x = -4.673358298 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + x = -4.673358298 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + x = -4.673358298 + -1.083333334 x = -4.673358298 + -1.083333334 Combine like terms: -4.673358298 + -1.083333334 = -5.756691632 x = -5.756691632 Simplifying x = -5.756691632Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.590024964, -5.756691632}
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