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Simplifying 12x2 + 5x + -32 = 0 Reorder the terms: -32 + 5x + 12x2 = 0 Solving -32 + 5x + 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'. -2.666666667 + 0.4166666667x + x2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + 0.4166666667x + 2.666666667 + x2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + 0.4166666667x + x2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + 0.4166666667x + x2 = 0 + 2.666666667 0.4166666667x + x2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 0.4166666667x + x2 = 2.666666667 The x term is 0.4166666667x. Take half its coefficient (0.2083333334). Square it (0.04340277781) and add it to both sides. Add '0.04340277781' to each side of the equation. 0.4166666667x + 0.04340277781 + x2 = 2.666666667 + 0.04340277781 Reorder the terms: 0.04340277781 + 0.4166666667x + x2 = 2.666666667 + 0.04340277781 Combine like terms: 2.666666667 + 0.04340277781 = 2.71006944481 0.04340277781 + 0.4166666667x + x2 = 2.71006944481 Factor a perfect square on the left side: (x + 0.2083333334)(x + 0.2083333334) = 2.71006944481 Calculate the square root of the right side: 1.646228856 Break this problem into two subproblems by setting (x + 0.2083333334) equal to 1.646228856 and -1.646228856.Subproblem 1
x + 0.2083333334 = 1.646228856 Simplifying x + 0.2083333334 = 1.646228856 Reorder the terms: 0.2083333334 + x = 1.646228856 Solving 0.2083333334 + x = 1.646228856 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2083333334' to each side of the equation. 0.2083333334 + -0.2083333334 + x = 1.646228856 + -0.2083333334 Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000 0.0000000000 + x = 1.646228856 + -0.2083333334 x = 1.646228856 + -0.2083333334 Combine like terms: 1.646228856 + -0.2083333334 = 1.4378955226 x = 1.4378955226 Simplifying x = 1.4378955226Subproblem 2
x + 0.2083333334 = -1.646228856 Simplifying x + 0.2083333334 = -1.646228856 Reorder the terms: 0.2083333334 + x = -1.646228856 Solving 0.2083333334 + x = -1.646228856 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2083333334' to each side of the equation. 0.2083333334 + -0.2083333334 + x = -1.646228856 + -0.2083333334 Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000 0.0000000000 + x = -1.646228856 + -0.2083333334 x = -1.646228856 + -0.2083333334 Combine like terms: -1.646228856 + -0.2083333334 = -1.8545621894 x = -1.8545621894 Simplifying x = -1.8545621894Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.4378955226, -1.8545621894}
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