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Simplifying -12x2 + 3x + 11 = 0 Reorder the terms: 11 + 3x + -12x2 = 0 Solving 11 + 3x + -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'. -0.9166666667 + -0.25x + x2 = 0 Move the constant term to the right: Add '0.9166666667' to each side of the equation. -0.9166666667 + -0.25x + 0.9166666667 + x2 = 0 + 0.9166666667 Reorder the terms: -0.9166666667 + 0.9166666667 + -0.25x + x2 = 0 + 0.9166666667 Combine like terms: -0.9166666667 + 0.9166666667 = 0.0000000000 0.0000000000 + -0.25x + x2 = 0 + 0.9166666667 -0.25x + x2 = 0 + 0.9166666667 Combine like terms: 0 + 0.9166666667 = 0.9166666667 -0.25x + x2 = 0.9166666667 The x term is -0.25x. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25x + 0.015625 + x2 = 0.9166666667 + 0.015625 Reorder the terms: 0.015625 + -0.25x + x2 = 0.9166666667 + 0.015625 Combine like terms: 0.9166666667 + 0.015625 = 0.9322916667 0.015625 + -0.25x + x2 = 0.9322916667 Factor a perfect square on the left side: (x + -0.125)(x + -0.125) = 0.9322916667 Calculate the square root of the right side: 0.965552519 Break this problem into two subproblems by setting (x + -0.125) equal to 0.965552519 and -0.965552519.Subproblem 1
x + -0.125 = 0.965552519 Simplifying x + -0.125 = 0.965552519 Reorder the terms: -0.125 + x = 0.965552519 Solving -0.125 + x = 0.965552519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = 0.965552519 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = 0.965552519 + 0.125 x = 0.965552519 + 0.125 Combine like terms: 0.965552519 + 0.125 = 1.090552519 x = 1.090552519 Simplifying x = 1.090552519Subproblem 2
x + -0.125 = -0.965552519 Simplifying x + -0.125 = -0.965552519 Reorder the terms: -0.125 + x = -0.965552519 Solving -0.125 + x = -0.965552519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = -0.965552519 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = -0.965552519 + 0.125 x = -0.965552519 + 0.125 Combine like terms: -0.965552519 + 0.125 = -0.840552519 x = -0.840552519 Simplifying x = -0.840552519Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.090552519, -0.840552519}
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