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Simplifying 32x2 + 54x + 7 = 0 Reorder the terms: 7 + 54x + 32x2 = 0 Solving 7 + 54x + 32x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. 0.21875 + 1.6875x + x2 = 0 Move the constant term to the right: Add '-0.21875' to each side of the equation. 0.21875 + 1.6875x + -0.21875 + x2 = 0 + -0.21875 Reorder the terms: 0.21875 + -0.21875 + 1.6875x + x2 = 0 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + 1.6875x + x2 = 0 + -0.21875 1.6875x + x2 = 0 + -0.21875 Combine like terms: 0 + -0.21875 = -0.21875 1.6875x + x2 = -0.21875 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 = -0.21875 + 0.7119140625 Reorder the terms: 0.7119140625 + 1.6875x + x2 = -0.21875 + 0.7119140625 Combine like terms: -0.21875 + 0.7119140625 = 0.4931640625 0.7119140625 + 1.6875x + x2 = 0.4931640625 Factor a perfect square on the left side: (x + 0.84375)(x + 0.84375) = 0.4931640625 Calculate the square root of the right side: 0.702256408 Break this problem into two subproblems by setting (x + 0.84375) equal to 0.702256408 and -0.702256408.Subproblem 1
x + 0.84375 = 0.702256408 Simplifying x + 0.84375 = 0.702256408 Reorder the terms: 0.84375 + x = 0.702256408 Solving 0.84375 + x = 0.702256408 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 = 0.702256408 + -0.84375 Combine like terms: 0.84375 + -0.84375 = 0.00000 0.00000 + x = 0.702256408 + -0.84375 x = 0.702256408 + -0.84375 Combine like terms: 0.702256408 + -0.84375 = -0.141493592 x = -0.141493592 Simplifying x = -0.141493592Subproblem 2
x + 0.84375 = -0.702256408 Simplifying x + 0.84375 = -0.702256408 Reorder the terms: 0.84375 + x = -0.702256408 Solving 0.84375 + x = -0.702256408 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 = -0.702256408 + -0.84375 Combine like terms: 0.84375 + -0.84375 = 0.00000 0.00000 + x = -0.702256408 + -0.84375 x = -0.702256408 + -0.84375 Combine like terms: -0.702256408 + -0.84375 = -1.546006408 x = -1.546006408 Simplifying x = -1.546006408Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.141493592, -1.546006408}
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