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Simplifying 2x(12 + x) = 75 (12 * 2x + x * 2x) = 75 (24x + 2x2) = 75 Solving 24x + 2x2 = 75 Solving for variable 'x'. Reorder the terms: -75 + 24x + 2x2 = 75 + -75 Combine like terms: 75 + -75 = 0 -75 + 24x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -37.5 + 12x + x2 = 0 Move the constant term to the right: Add '37.5' to each side of the equation. -37.5 + 12x + 37.5 + x2 = 0 + 37.5 Reorder the terms: -37.5 + 37.5 + 12x + x2 = 0 + 37.5 Combine like terms: -37.5 + 37.5 = 0.0 0.0 + 12x + x2 = 0 + 37.5 12x + x2 = 0 + 37.5 Combine like terms: 0 + 37.5 = 37.5 12x + x2 = 37.5 The x term is 12x. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12x + 36 + x2 = 37.5 + 36 Reorder the terms: 36 + 12x + x2 = 37.5 + 36 Combine like terms: 37.5 + 36 = 73.5 36 + 12x + x2 = 73.5 Factor a perfect square on the left side: (x + 6)(x + 6) = 73.5 Calculate the square root of the right side: 8.5732141 Break this problem into two subproblems by setting (x + 6) equal to 8.5732141 and -8.5732141.Subproblem 1
x + 6 = 8.5732141 Simplifying x + 6 = 8.5732141 Reorder the terms: 6 + x = 8.5732141 Solving 6 + x = 8.5732141 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = 8.5732141 + -6 Combine like terms: 6 + -6 = 0 0 + x = 8.5732141 + -6 x = 8.5732141 + -6 Combine like terms: 8.5732141 + -6 = 2.5732141 x = 2.5732141 Simplifying x = 2.5732141Subproblem 2
x + 6 = -8.5732141 Simplifying x + 6 = -8.5732141 Reorder the terms: 6 + x = -8.5732141 Solving 6 + x = -8.5732141 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = -8.5732141 + -6 Combine like terms: 6 + -6 = 0 0 + x = -8.5732141 + -6 x = -8.5732141 + -6 Combine like terms: -8.5732141 + -6 = -14.5732141 x = -14.5732141 Simplifying x = -14.5732141Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.5732141, -14.5732141}
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