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Simplifying 0 = -4X2 + 3X + 3 Reorder the terms: 0 = 3 + 3X + -4X2 Solving 0 = 3 + 3X + -4X2 Solving for variable 'X'. Combine like terms: 0 + -3 = -3 -3 + -3X + 4X2 = 3 + 3X + -4X2 + -3 + -3X + 4X2 Reorder the terms: -3 + -3X + 4X2 = 3 + -3 + 3X + -3X + -4X2 + 4X2 Combine like terms: 3 + -3 = 0 -3 + -3X + 4X2 = 0 + 3X + -3X + -4X2 + 4X2 -3 + -3X + 4X2 = 3X + -3X + -4X2 + 4X2 Combine like terms: 3X + -3X = 0 -3 + -3X + 4X2 = 0 + -4X2 + 4X2 -3 + -3X + 4X2 = -4X2 + 4X2 Combine like terms: -4X2 + 4X2 = 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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