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Simplifying -4x2 + 11x = 3 Reorder the terms: 11x + -4x2 = 3 Solving 11x + -4x2 = 3 Solving for variable 'x'. Reorder the terms: -3 + 11x + -4x2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 11x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 0.75 + -2.75x + x2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2.75x + -0.75 + x2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2.75x + x2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2.75x + x2 = 0 + -0.75 -2.75x + x2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2.75x + x2 = -0.75 The x term is -2.75x. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75x + 1.890625 + x2 = -0.75 + 1.890625 Reorder the terms: 1.890625 + -2.75x + x2 = -0.75 + 1.890625 Combine like terms: -0.75 + 1.890625 = 1.140625 1.890625 + -2.75x + x2 = 1.140625 Factor a perfect square on the left side: (x + -1.375)(x + -1.375) = 1.140625 Calculate the square root of the right side: 1.068000468 Break this problem into two subproblems by setting (x + -1.375) equal to 1.068000468 and -1.068000468.Subproblem 1
x + -1.375 = 1.068000468 Simplifying x + -1.375 = 1.068000468 Reorder the terms: -1.375 + x = 1.068000468 Solving -1.375 + x = 1.068000468 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = 1.068000468 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = 1.068000468 + 1.375 x = 1.068000468 + 1.375 Combine like terms: 1.068000468 + 1.375 = 2.443000468 x = 2.443000468 Simplifying x = 2.443000468Subproblem 2
x + -1.375 = -1.068000468 Simplifying x + -1.375 = -1.068000468 Reorder the terms: -1.375 + x = -1.068000468 Solving -1.375 + x = -1.068000468 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = -1.068000468 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = -1.068000468 + 1.375 x = -1.068000468 + 1.375 Combine like terms: -1.068000468 + 1.375 = 0.306999532 x = 0.306999532 Simplifying x = 0.306999532Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.443000468, 0.306999532}
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