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Simplifying (2x + 3.75) * x + -0.5 = 11.5 Reorder the terms: (3.75 + 2x) * x + -0.5 = 11.5 Reorder the terms for easier multiplication: x(3.75 + 2x) + -0.5 = 11.5 (3.75 * x + 2x * x) + -0.5 = 11.5 (3.75x + 2x2) + -0.5 = 11.5 Reorder the terms: -0.5 + 3.75x + 2x2 = 11.5 Solving -0.5 + 3.75x + 2x2 = 11.5 Solving for variable 'x'. Reorder the terms: -0.5 + -11.5 + 3.75x + 2x2 = 11.5 + -11.5 Combine like terms: -0.5 + -11.5 = -12 -12 + 3.75x + 2x2 = 11.5 + -11.5 Combine like terms: 11.5 + -11.5 = 0.0 -12 + 3.75x + 2x2 = 0.0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6 + 1.875x + x2 = 0.0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 1.875x + 6 + x2 = 0.0 + 6 Reorder the terms: -6 + 6 + 1.875x + x2 = 0.0 + 6 Combine like terms: -6 + 6 = 0 0 + 1.875x + x2 = 0.0 + 6 1.875x + x2 = 0.0 + 6 Combine like terms: 0.0 + 6 = 6 1.875x + x2 = 6 The x term is 1.875x. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875x + 0.87890625 + x2 = 6 + 0.87890625 Reorder the terms: 0.87890625 + 1.875x + x2 = 6 + 0.87890625 Combine like terms: 6 + 0.87890625 = 6.87890625 0.87890625 + 1.875x + x2 = 6.87890625 Factor a perfect square on the left side: (x + 0.9375)(x + 0.9375) = 6.87890625 Calculate the square root of the right side: 2.622766907 Break this problem into two subproblems by setting (x + 0.9375) equal to 2.622766907 and -2.622766907.Subproblem 1
x + 0.9375 = 2.622766907 Simplifying x + 0.9375 = 2.622766907 Reorder the terms: 0.9375 + x = 2.622766907 Solving 0.9375 + x = 2.622766907 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + x = 2.622766907 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + x = 2.622766907 + -0.9375 x = 2.622766907 + -0.9375 Combine like terms: 2.622766907 + -0.9375 = 1.685266907 x = 1.685266907 Simplifying x = 1.685266907Subproblem 2
x + 0.9375 = -2.622766907 Simplifying x + 0.9375 = -2.622766907 Reorder the terms: 0.9375 + x = -2.622766907 Solving 0.9375 + x = -2.622766907 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + x = -2.622766907 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + x = -2.622766907 + -0.9375 x = -2.622766907 + -0.9375 Combine like terms: -2.622766907 + -0.9375 = -3.560266907 x = -3.560266907 Simplifying x = -3.560266907Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.685266907, -3.560266907}
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