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Simplifying 6 = -16x2 + 27x + 5 Reorder the terms: 6 = 5 + 27x + -16x2 Solving 6 = 5 + 27x + -16x2 Solving for variable 'x'. Combine like terms: 6 + -5 = 1 1 + -27x + 16x2 = 5 + 27x + -16x2 + -5 + -27x + 16x2 Reorder the terms: 1 + -27x + 16x2 = 5 + -5 + 27x + -27x + -16x2 + 16x2 Combine like terms: 5 + -5 = 0 1 + -27x + 16x2 = 0 + 27x + -27x + -16x2 + 16x2 1 + -27x + 16x2 = 27x + -27x + -16x2 + 16x2 Combine like terms: 27x + -27x = 0 1 + -27x + 16x2 = 0 + -16x2 + 16x2 1 + -27x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 1 + -27x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.0625 + -1.6875x + x2 = 0 Move the constant term to the right: Add '-0.0625' to each side of the equation. 0.0625 + -1.6875x + -0.0625 + x2 = 0 + -0.0625 Reorder the terms: 0.0625 + -0.0625 + -1.6875x + x2 = 0 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + -1.6875x + x2 = 0 + -0.0625 -1.6875x + x2 = 0 + -0.0625 Combine like terms: 0 + -0.0625 = -0.0625 -1.6875x + x2 = -0.0625 The x term is -1.6875x. Take half its coefficient (-0.84375). Square it (0.7119140625) and add it to both sides. Add '0.7119140625' to each side of the equation. -1.6875x + 0.7119140625 + x2 = -0.0625 + 0.7119140625 Reorder the terms: 0.7119140625 + -1.6875x + x2 = -0.0625 + 0.7119140625 Combine like terms: -0.0625 + 0.7119140625 = 0.6494140625 0.7119140625 + -1.6875x + x2 = 0.6494140625 Factor a perfect square on the left side: (x + -0.84375)(x + -0.84375) = 0.6494140625 Calculate the square root of the right side: 0.80586231 Break this problem into two subproblems by setting (x + -0.84375) equal to 0.80586231 and -0.80586231.Subproblem 1
x + -0.84375 = 0.80586231 Simplifying x + -0.84375 = 0.80586231 Reorder the terms: -0.84375 + x = 0.80586231 Solving -0.84375 + x = 0.80586231 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = 0.80586231 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = 0.80586231 + 0.84375 x = 0.80586231 + 0.84375 Combine like terms: 0.80586231 + 0.84375 = 1.64961231 x = 1.64961231 Simplifying x = 1.64961231Subproblem 2
x + -0.84375 = -0.80586231 Simplifying x + -0.84375 = -0.80586231 Reorder the terms: -0.84375 + x = -0.80586231 Solving -0.84375 + x = -0.80586231 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.84375' to each side of the equation. -0.84375 + 0.84375 + x = -0.80586231 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + x = -0.80586231 + 0.84375 x = -0.80586231 + 0.84375 Combine like terms: -0.80586231 + 0.84375 = 0.03788769 x = 0.03788769 Simplifying x = 0.03788769Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.64961231, 0.03788769}
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