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Simplifying 27x2 + -48x + 11 = 2 Reorder the terms: 11 + -48x + 27x2 = 2 Solving 11 + -48x + 27x2 = 2 Solving for variable 'x'. Reorder the terms: 11 + -2 + -48x + 27x2 = 2 + -2 Combine like terms: 11 + -2 = 9 9 + -48x + 27x2 = 2 + -2 Combine like terms: 2 + -2 = 0 9 + -48x + 27x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(3 + -16x + 9x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(3 + -16x + 9x2)' equal to zero and attempt to solve: Simplifying 3 + -16x + 9x2 = 0 Solving 3 + -16x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.3333333333 + -1.777777778x + x2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -1.777777778x + -0.3333333333 + x2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -1.777777778x + x2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -1.777777778x + x2 = 0 + -0.3333333333 -1.777777778x + x2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -1.777777778x + x2 = -0.3333333333 The x term is -1.777777778x. Take half its coefficient (-0.888888889). Square it (0.7901234570) and add it to both sides. Add '0.7901234570' to each side of the equation. -1.777777778x + 0.7901234570 + x2 = -0.3333333333 + 0.7901234570 Reorder the terms: 0.7901234570 + -1.777777778x + x2 = -0.3333333333 + 0.7901234570 Combine like terms: -0.3333333333 + 0.7901234570 = 0.4567901237 0.7901234570 + -1.777777778x + x2 = 0.4567901237 Factor a perfect square on the left side: (x + -0.888888889)(x + -0.888888889) = 0.4567901237 Calculate the square root of the right side: 0.675862504 Break this problem into two subproblems by setting (x + -0.888888889) equal to 0.675862504 and -0.675862504.Subproblem 1
x + -0.888888889 = 0.675862504 Simplifying x + -0.888888889 = 0.675862504 Reorder the terms: -0.888888889 + x = 0.675862504 Solving -0.888888889 + x = 0.675862504 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.888888889' to each side of the equation. -0.888888889 + 0.888888889 + x = 0.675862504 + 0.888888889 Combine like terms: -0.888888889 + 0.888888889 = 0.000000000 0.000000000 + x = 0.675862504 + 0.888888889 x = 0.675862504 + 0.888888889 Combine like terms: 0.675862504 + 0.888888889 = 1.564751393 x = 1.564751393 Simplifying x = 1.564751393Subproblem 2
x + -0.888888889 = -0.675862504 Simplifying x + -0.888888889 = -0.675862504 Reorder the terms: -0.888888889 + x = -0.675862504 Solving -0.888888889 + x = -0.675862504 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.888888889' to each side of the equation. -0.888888889 + 0.888888889 + x = -0.675862504 + 0.888888889 Combine like terms: -0.888888889 + 0.888888889 = 0.000000000 0.000000000 + x = -0.675862504 + 0.888888889 x = -0.675862504 + 0.888888889 Combine like terms: -0.675862504 + 0.888888889 = 0.213026385 x = 0.213026385 Simplifying x = 0.213026385Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.564751393, 0.213026385}Solution
x = {1.564751393, 0.213026385}
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