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Simplifying 27x2 + 90x + -60 = 0 Reorder the terms: -60 + 90x + 27x2 = 0 Solving -60 + 90x + 27x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(-20 + 30x + 9x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-20 + 30x + 9x2)' equal to zero and attempt to solve: Simplifying -20 + 30x + 9x2 = 0 Solving -20 + 30x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.222222222 + 3.333333333x + x2 = 0 Move the constant term to the right: Add '2.222222222' to each side of the equation. -2.222222222 + 3.333333333x + 2.222222222 + x2 = 0 + 2.222222222 Reorder the terms: -2.222222222 + 2.222222222 + 3.333333333x + x2 = 0 + 2.222222222 Combine like terms: -2.222222222 + 2.222222222 = 0.000000000 0.000000000 + 3.333333333x + x2 = 0 + 2.222222222 3.333333333x + x2 = 0 + 2.222222222 Combine like terms: 0 + 2.222222222 = 2.222222222 3.333333333x + x2 = 2.222222222 The x term is 3.333333333x. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333x + 2.777777779 + x2 = 2.222222222 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333x + x2 = 2.222222222 + 2.777777779 Combine like terms: 2.222222222 + 2.777777779 = 5.000000001 2.777777779 + 3.333333333x + x2 = 5.000000001 Factor a perfect square on the left side: (x + 1.666666667)(x + 1.666666667) = 5.000000001 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (x + 1.666666667) equal to 2.236067978 and -2.236067978.Subproblem 1
x + 1.666666667 = 2.236067978 Simplifying x + 1.666666667 = 2.236067978 Reorder the terms: 1.666666667 + x = 2.236067978 Solving 1.666666667 + x = 2.236067978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = 2.236067978 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = 2.236067978 + -1.666666667 x = 2.236067978 + -1.666666667 Combine like terms: 2.236067978 + -1.666666667 = 0.569401311 x = 0.569401311 Simplifying x = 0.569401311Subproblem 2
x + 1.666666667 = -2.236067978 Simplifying x + 1.666666667 = -2.236067978 Reorder the terms: 1.666666667 + x = -2.236067978 Solving 1.666666667 + x = -2.236067978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = -2.236067978 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = -2.236067978 + -1.666666667 x = -2.236067978 + -1.666666667 Combine like terms: -2.236067978 + -1.666666667 = -3.902734645 x = -3.902734645 Simplifying x = -3.902734645Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.569401311, -3.902734645}Solution
x = {0.569401311, -3.902734645}
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