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Simplifying 12x2 + 27x + 2 = 0 Reorder the terms: 2 + 27x + 12x2 = 0 Solving 2 + 27x + 12x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.1666666667 + 2.25x + x2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + 2.25x + -0.1666666667 + x2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + 2.25x + x2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + 2.25x + x2 = 0 + -0.1666666667 2.25x + x2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 2.25x + x2 = -0.1666666667 The x term is 2.25x. Take half its coefficient (1.125). Square it (1.265625) and add it to both sides. Add '1.265625' to each side of the equation. 2.25x + 1.265625 + x2 = -0.1666666667 + 1.265625 Reorder the terms: 1.265625 + 2.25x + x2 = -0.1666666667 + 1.265625 Combine like terms: -0.1666666667 + 1.265625 = 1.0989583333 1.265625 + 2.25x + x2 = 1.0989583333 Factor a perfect square on the left side: (x + 1.125)(x + 1.125) = 1.0989583333 Calculate the square root of the right side: 1.048312135 Break this problem into two subproblems by setting (x + 1.125) equal to 1.048312135 and -1.048312135.Subproblem 1
x + 1.125 = 1.048312135 Simplifying x + 1.125 = 1.048312135 Reorder the terms: 1.125 + x = 1.048312135 Solving 1.125 + x = 1.048312135 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = 1.048312135 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = 1.048312135 + -1.125 x = 1.048312135 + -1.125 Combine like terms: 1.048312135 + -1.125 = -0.076687865 x = -0.076687865 Simplifying x = -0.076687865Subproblem 2
x + 1.125 = -1.048312135 Simplifying x + 1.125 = -1.048312135 Reorder the terms: 1.125 + x = -1.048312135 Solving 1.125 + x = -1.048312135 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + x = -1.048312135 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + x = -1.048312135 + -1.125 x = -1.048312135 + -1.125 Combine like terms: -1.048312135 + -1.125 = -2.173312135 x = -2.173312135 Simplifying x = -2.173312135Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.076687865, -2.173312135}
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