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Simplifying 32x2 + -9x + -33 = 0 Reorder the terms: -33 + -9x + 32x2 = 0 Solving -33 + -9x + 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'. -1.03125 + -0.28125x + x2 = 0 Move the constant term to the right: Add '1.03125' to each side of the equation. -1.03125 + -0.28125x + 1.03125 + x2 = 0 + 1.03125 Reorder the terms: -1.03125 + 1.03125 + -0.28125x + x2 = 0 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + -0.28125x + x2 = 0 + 1.03125 -0.28125x + x2 = 0 + 1.03125 Combine like terms: 0 + 1.03125 = 1.03125 -0.28125x + x2 = 1.03125 The x term is -0.28125x. Take half its coefficient (-0.140625). Square it (0.01977539063) and add it to both sides. Add '0.01977539063' to each side of the equation. -0.28125x + 0.01977539063 + x2 = 1.03125 + 0.01977539063 Reorder the terms: 0.01977539063 + -0.28125x + x2 = 1.03125 + 0.01977539063 Combine like terms: 1.03125 + 0.01977539063 = 1.05102539063 0.01977539063 + -0.28125x + x2 = 1.05102539063 Factor a perfect square on the left side: (x + -0.140625)(x + -0.140625) = 1.05102539063 Calculate the square root of the right side: 1.025195294 Break this problem into two subproblems by setting (x + -0.140625) equal to 1.025195294 and -1.025195294.Subproblem 1
x + -0.140625 = 1.025195294 Simplifying x + -0.140625 = 1.025195294 Reorder the terms: -0.140625 + x = 1.025195294 Solving -0.140625 + x = 1.025195294 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.140625' to each side of the equation. -0.140625 + 0.140625 + x = 1.025195294 + 0.140625 Combine like terms: -0.140625 + 0.140625 = 0.000000 0.000000 + x = 1.025195294 + 0.140625 x = 1.025195294 + 0.140625 Combine like terms: 1.025195294 + 0.140625 = 1.165820294 x = 1.165820294 Simplifying x = 1.165820294Subproblem 2
x + -0.140625 = -1.025195294 Simplifying x + -0.140625 = -1.025195294 Reorder the terms: -0.140625 + x = -1.025195294 Solving -0.140625 + x = -1.025195294 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.140625' to each side of the equation. -0.140625 + 0.140625 + x = -1.025195294 + 0.140625 Combine like terms: -0.140625 + 0.140625 = 0.000000 0.000000 + x = -1.025195294 + 0.140625 x = -1.025195294 + 0.140625 Combine like terms: -1.025195294 + 0.140625 = -0.884570294 x = -0.884570294 Simplifying x = -0.884570294Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.165820294, -0.884570294}
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