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Simplifying -3S(2S + -4) + -12 = -3(5S + 3) + -4 Reorder the terms: -3S(-4 + 2S) + -12 = -3(5S + 3) + -4 (-4 * -3S + 2S * -3S) + -12 = -3(5S + 3) + -4 (12S + -6S2) + -12 = -3(5S + 3) + -4 Reorder the terms: -12 + 12S + -6S2 = -3(5S + 3) + -4 Reorder the terms: -12 + 12S + -6S2 = -3(3 + 5S) + -4 -12 + 12S + -6S2 = (3 * -3 + 5S * -3) + -4 -12 + 12S + -6S2 = (-9 + -15S) + -4 Reorder the terms: -12 + 12S + -6S2 = -9 + -4 + -15S Combine like terms: -9 + -4 = -13 -12 + 12S + -6S2 = -13 + -15S Solving -12 + 12S + -6S2 = -13 + -15S Solving for variable 'S'. Reorder the terms: -12 + 13 + 12S + 15S + -6S2 = -13 + -15S + 13 + 15S Combine like terms: -12 + 13 = 1 1 + 12S + 15S + -6S2 = -13 + -15S + 13 + 15S Combine like terms: 12S + 15S = 27S 1 + 27S + -6S2 = -13 + -15S + 13 + 15S Reorder the terms: 1 + 27S + -6S2 = -13 + 13 + -15S + 15S Combine like terms: -13 + 13 = 0 1 + 27S + -6S2 = 0 + -15S + 15S 1 + 27S + -6S2 = -15S + 15S Combine like terms: -15S + 15S = 0 1 + 27S + -6S2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -0.1666666667 + -4.5S + S2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + -4.5S + 0.1666666667 + S2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + -4.5S + S2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + -4.5S + S2 = 0 + 0.1666666667 -4.5S + S2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 -4.5S + S2 = 0.1666666667 The S term is -4.5S. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5S + 5.0625 + S2 = 0.1666666667 + 5.0625 Reorder the terms: 5.0625 + -4.5S + S2 = 0.1666666667 + 5.0625 Combine like terms: 0.1666666667 + 5.0625 = 5.2291666667 5.0625 + -4.5S + S2 = 5.2291666667 Factor a perfect square on the left side: (S + -2.25)(S + -2.25) = 5.2291666667 Calculate the square root of the right side: 2.286737122 Break this problem into two subproblems by setting (S + -2.25) equal to 2.286737122 and -2.286737122.Subproblem 1
S + -2.25 = 2.286737122 Simplifying S + -2.25 = 2.286737122 Reorder the terms: -2.25 + S = 2.286737122 Solving -2.25 + S = 2.286737122 Solving for variable 'S'. Move all terms containing S to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + S = 2.286737122 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + S = 2.286737122 + 2.25 S = 2.286737122 + 2.25 Combine like terms: 2.286737122 + 2.25 = 4.536737122 S = 4.536737122 Simplifying S = 4.536737122Subproblem 2
S + -2.25 = -2.286737122 Simplifying S + -2.25 = -2.286737122 Reorder the terms: -2.25 + S = -2.286737122 Solving -2.25 + S = -2.286737122 Solving for variable 'S'. Move all terms containing S to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + S = -2.286737122 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + S = -2.286737122 + 2.25 S = -2.286737122 + 2.25 Combine like terms: -2.286737122 + 2.25 = -0.036737122 S = -0.036737122 Simplifying S = -0.036737122Solution
The solution to the problem is based on the solutions from the subproblems. S = {4.536737122, -0.036737122}
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