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Simplifying 4x(x + 1) = 3 + 7x Reorder the terms: 4x(1 + x) = 3 + 7x (1 * 4x + x * 4x) = 3 + 7x (4x + 4x2) = 3 + 7x Solving 4x + 4x2 = 3 + 7x Solving for variable 'x'. Reorder the terms: -3 + 4x + -7x + 4x2 = 3 + 7x + -3 + -7x Combine like terms: 4x + -7x = -3x -3 + -3x + 4x2 = 3 + 7x + -3 + -7x Reorder the terms: -3 + -3x + 4x2 = 3 + -3 + 7x + -7x Combine like terms: 3 + -3 = 0 -3 + -3x + 4x2 = 0 + 7x + -7x -3 + -3x + 4x2 = 7x + -7x Combine like terms: 7x + -7x = 0 -3 + -3x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -0.75x + x2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.75x + 0.75 + x2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.75x + x2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.75x + x2 = 0 + 0.75 -0.75x + x2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.75x + x2 = 0.75 The x term is -0.75x. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75x + 0.140625 + x2 = 0.75 + 0.140625 Reorder the terms: 0.140625 + -0.75x + x2 = 0.75 + 0.140625 Combine like terms: 0.75 + 0.140625 = 0.890625 0.140625 + -0.75x + x2 = 0.890625 Factor a perfect square on the left side: (x + -0.375)(x + -0.375) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (x + -0.375) equal to 0.943729304 and -0.943729304.Subproblem 1
x + -0.375 = 0.943729304 Simplifying x + -0.375 = 0.943729304 Reorder the terms: -0.375 + x = 0.943729304 Solving -0.375 + x = 0.943729304 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 0.943729304 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 0.943729304 + 0.375 x = 0.943729304 + 0.375 Combine like terms: 0.943729304 + 0.375 = 1.318729304 x = 1.318729304 Simplifying x = 1.318729304Subproblem 2
x + -0.375 = -0.943729304 Simplifying x + -0.375 = -0.943729304 Reorder the terms: -0.375 + x = -0.943729304 Solving -0.375 + x = -0.943729304 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -0.943729304 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -0.943729304 + 0.375 x = -0.943729304 + 0.375 Combine like terms: -0.943729304 + 0.375 = -0.568729304 x = -0.568729304 Simplifying x = -0.568729304Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.318729304, -0.568729304}
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