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Simplifying 0 = -4x2 + -0.25x + 1.225 Reorder the terms: 0 = 1.225 + -0.25x + -4x2 Solving 0 = 1.225 + -0.25x + -4x2 Solving for variable 'x'. Combine like terms: 0 + -1.225 = -1.225 -1.225 + 0.25x + 4x2 = 1.225 + -0.25x + -4x2 + -1.225 + 0.25x + 4x2 Reorder the terms: -1.225 + 0.25x + 4x2 = 1.225 + -1.225 + -0.25x + 0.25x + -4x2 + 4x2 Combine like terms: 1.225 + -1.225 = 0.000 -1.225 + 0.25x + 4x2 = 0.000 + -0.25x + 0.25x + -4x2 + 4x2 -1.225 + 0.25x + 4x2 = -0.25x + 0.25x + -4x2 + 4x2 Combine like terms: -0.25x + 0.25x = 0.00 -1.225 + 0.25x + 4x2 = 0.00 + -4x2 + 4x2 -1.225 + 0.25x + 4x2 = -4x2 + 4x2 Combine like terms: -4x2 + 4x2 = 0 -1.225 + 0.25x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.30625 + 0.0625x + x2 = 0 Move the constant term to the right: Add '0.30625' to each side of the equation. -0.30625 + 0.0625x + 0.30625 + x2 = 0 + 0.30625 Reorder the terms: -0.30625 + 0.30625 + 0.0625x + x2 = 0 + 0.30625 Combine like terms: -0.30625 + 0.30625 = 0.00000 0.00000 + 0.0625x + x2 = 0 + 0.30625 0.0625x + x2 = 0 + 0.30625 Combine like terms: 0 + 0.30625 = 0.30625 0.0625x + x2 = 0.30625 The x term is 0.0625x. Take half its coefficient (0.03125). Square it (0.0009765625) and add it to both sides. Add '0.0009765625' to each side of the equation. 0.0625x + 0.0009765625 + x2 = 0.30625 + 0.0009765625 Reorder the terms: 0.0009765625 + 0.0625x + x2 = 0.30625 + 0.0009765625 Combine like terms: 0.30625 + 0.0009765625 = 0.3072265625 0.0009765625 + 0.0625x + x2 = 0.3072265625 Factor a perfect square on the left side: (x + 0.03125)(x + 0.03125) = 0.3072265625 Calculate the square root of the right side: 0.55428022 Break this problem into two subproblems by setting (x + 0.03125) equal to 0.55428022 and -0.55428022.Subproblem 1
x + 0.03125 = 0.55428022 Simplifying x + 0.03125 = 0.55428022 Reorder the terms: 0.03125 + x = 0.55428022 Solving 0.03125 + x = 0.55428022 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.03125' to each side of the equation. 0.03125 + -0.03125 + x = 0.55428022 + -0.03125 Combine like terms: 0.03125 + -0.03125 = 0.00000 0.00000 + x = 0.55428022 + -0.03125 x = 0.55428022 + -0.03125 Combine like terms: 0.55428022 + -0.03125 = 0.52303022 x = 0.52303022 Simplifying x = 0.52303022Subproblem 2
x + 0.03125 = -0.55428022 Simplifying x + 0.03125 = -0.55428022 Reorder the terms: 0.03125 + x = -0.55428022 Solving 0.03125 + x = -0.55428022 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.03125' to each side of the equation. 0.03125 + -0.03125 + x = -0.55428022 + -0.03125 Combine like terms: 0.03125 + -0.03125 = 0.00000 0.00000 + x = -0.55428022 + -0.03125 x = -0.55428022 + -0.03125 Combine like terms: -0.55428022 + -0.03125 = -0.58553022 x = -0.58553022 Simplifying x = -0.58553022Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.52303022, -0.58553022}
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