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Simplifying 32x2 + 40x + 9 = 0 Reorder the terms: 9 + 40x + 32x2 = 0 Solving 9 + 40x + 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'. 0.28125 + 1.25x + x2 = 0 Move the constant term to the right: Add '-0.28125' to each side of the equation. 0.28125 + 1.25x + -0.28125 + x2 = 0 + -0.28125 Reorder the terms: 0.28125 + -0.28125 + 1.25x + x2 = 0 + -0.28125 Combine like terms: 0.28125 + -0.28125 = 0.00000 0.00000 + 1.25x + x2 = 0 + -0.28125 1.25x + x2 = 0 + -0.28125 Combine like terms: 0 + -0.28125 = -0.28125 1.25x + x2 = -0.28125 The x term is 1.25x. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25x + 0.390625 + x2 = -0.28125 + 0.390625 Reorder the terms: 0.390625 + 1.25x + x2 = -0.28125 + 0.390625 Combine like terms: -0.28125 + 0.390625 = 0.109375 0.390625 + 1.25x + x2 = 0.109375 Factor a perfect square on the left side: (x + 0.625)(x + 0.625) = 0.109375 Calculate the square root of the right side: 0.330718914 Break this problem into two subproblems by setting (x + 0.625) equal to 0.330718914 and -0.330718914.Subproblem 1
x + 0.625 = 0.330718914 Simplifying x + 0.625 = 0.330718914 Reorder the terms: 0.625 + x = 0.330718914 Solving 0.625 + x = 0.330718914 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 0.330718914 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 0.330718914 + -0.625 x = 0.330718914 + -0.625 Combine like terms: 0.330718914 + -0.625 = -0.294281086 x = -0.294281086 Simplifying x = -0.294281086Subproblem 2
x + 0.625 = -0.330718914 Simplifying x + 0.625 = -0.330718914 Reorder the terms: 0.625 + x = -0.330718914 Solving 0.625 + x = -0.330718914 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -0.330718914 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -0.330718914 + -0.625 x = -0.330718914 + -0.625 Combine like terms: -0.330718914 + -0.625 = -0.955718914 x = -0.955718914 Simplifying x = -0.955718914Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.294281086, -0.955718914}
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