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Simplifying 64x2 + 27x + -64 = 0 Reorder the terms: -64 + 27x + 64x2 = 0 Solving -64 + 27x + 64x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. -1 + 0.421875x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.421875x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 0.421875x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.421875x + x2 = 0 + 1 0.421875x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.421875x + x2 = 1 The x term is 0.421875x. Take half its coefficient (0.2109375). Square it (0.04449462891) and add it to both sides. Add '0.04449462891' to each side of the equation. 0.421875x + 0.04449462891 + x2 = 1 + 0.04449462891 Reorder the terms: 0.04449462891 + 0.421875x + x2 = 1 + 0.04449462891 Combine like terms: 1 + 0.04449462891 = 1.04449462891 0.04449462891 + 0.421875x + x2 = 1.04449462891 Factor a perfect square on the left side: (x + 0.2109375)(x + 0.2109375) = 1.04449462891 Calculate the square root of the right side: 1.0220052 Break this problem into two subproblems by setting (x + 0.2109375) equal to 1.0220052 and -1.0220052.Subproblem 1
x + 0.2109375 = 1.0220052 Simplifying x + 0.2109375 = 1.0220052 Reorder the terms: 0.2109375 + x = 1.0220052 Solving 0.2109375 + x = 1.0220052 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2109375' to each side of the equation. 0.2109375 + -0.2109375 + x = 1.0220052 + -0.2109375 Combine like terms: 0.2109375 + -0.2109375 = 0.0000000 0.0000000 + x = 1.0220052 + -0.2109375 x = 1.0220052 + -0.2109375 Combine like terms: 1.0220052 + -0.2109375 = 0.8110677 x = 0.8110677 Simplifying x = 0.8110677Subproblem 2
x + 0.2109375 = -1.0220052 Simplifying x + 0.2109375 = -1.0220052 Reorder the terms: 0.2109375 + x = -1.0220052 Solving 0.2109375 + x = -1.0220052 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2109375' to each side of the equation. 0.2109375 + -0.2109375 + x = -1.0220052 + -0.2109375 Combine like terms: 0.2109375 + -0.2109375 = 0.0000000 0.0000000 + x = -1.0220052 + -0.2109375 x = -1.0220052 + -0.2109375 Combine like terms: -1.0220052 + -0.2109375 = -1.2329427 x = -1.2329427 Simplifying x = -1.2329427Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8110677, -1.2329427}
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