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Simplifying 3x = (9x + 1)(27 + -2x) Reorder the terms: 3x = (1 + 9x)(27 + -2x) Multiply (1 + 9x) * (27 + -2x) 3x = (1(27 + -2x) + 9x * (27 + -2x)) 3x = ((27 * 1 + -2x * 1) + 9x * (27 + -2x)) 3x = ((27 + -2x) + 9x * (27 + -2x)) 3x = (27 + -2x + (27 * 9x + -2x * 9x)) 3x = (27 + -2x + (243x + -18x2)) Combine like terms: -2x + 243x = 241x 3x = (27 + 241x + -18x2) Solving 3x = 27 + 241x + -18x2 Solving for variable 'x'. Reorder the terms: -27 + 3x + -241x + 18x2 = 27 + 241x + -18x2 + -27 + -241x + 18x2 Combine like terms: 3x + -241x = -238x -27 + -238x + 18x2 = 27 + 241x + -18x2 + -27 + -241x + 18x2 Reorder the terms: -27 + -238x + 18x2 = 27 + -27 + 241x + -241x + -18x2 + 18x2 Combine like terms: 27 + -27 = 0 -27 + -238x + 18x2 = 0 + 241x + -241x + -18x2 + 18x2 -27 + -238x + 18x2 = 241x + -241x + -18x2 + 18x2 Combine like terms: 241x + -241x = 0 -27 + -238x + 18x2 = 0 + -18x2 + 18x2 -27 + -238x + 18x2 = -18x2 + 18x2 Combine like terms: -18x2 + 18x2 = 0 -27 + -238x + 18x2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -1.5 + -13.22222222x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -13.22222222x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -13.22222222x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -13.22222222x + x2 = 0 + 1.5 -13.22222222x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -13.22222222x + x2 = 1.5 The x term is -13.22222222x. Take half its coefficient (-6.61111111). Square it (43.70679011) and add it to both sides. Add '43.70679011' to each side of the equation. -13.22222222x + 43.70679011 + x2 = 1.5 + 43.70679011 Reorder the terms: 43.70679011 + -13.22222222x + x2 = 1.5 + 43.70679011 Combine like terms: 1.5 + 43.70679011 = 45.20679011 43.70679011 + -13.22222222x + x2 = 45.20679011 Factor a perfect square on the left side: (x + -6.61111111)(x + -6.61111111) = 45.20679011 Calculate the square root of the right side: 6.723599491 Break this problem into two subproblems by setting (x + -6.61111111) equal to 6.723599491 and -6.723599491.Subproblem 1
x + -6.61111111 = 6.723599491 Simplifying x + -6.61111111 = 6.723599491 Reorder the terms: -6.61111111 + x = 6.723599491 Solving -6.61111111 + x = 6.723599491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.61111111' to each side of the equation. -6.61111111 + 6.61111111 + x = 6.723599491 + 6.61111111 Combine like terms: -6.61111111 + 6.61111111 = 0.00000000 0.00000000 + x = 6.723599491 + 6.61111111 x = 6.723599491 + 6.61111111 Combine like terms: 6.723599491 + 6.61111111 = 13.334710601 x = 13.334710601 Simplifying x = 13.334710601Subproblem 2
x + -6.61111111 = -6.723599491 Simplifying x + -6.61111111 = -6.723599491 Reorder the terms: -6.61111111 + x = -6.723599491 Solving -6.61111111 + x = -6.723599491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.61111111' to each side of the equation. -6.61111111 + 6.61111111 + x = -6.723599491 + 6.61111111 Combine like terms: -6.61111111 + 6.61111111 = 0.00000000 0.00000000 + x = -6.723599491 + 6.61111111 x = -6.723599491 + 6.61111111 Combine like terms: -6.723599491 + 6.61111111 = -0.112488381 x = -0.112488381 Simplifying x = -0.112488381Solution
The solution to the problem is based on the solutions from the subproblems. x = {13.334710601, -0.112488381}
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