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Simplifying 0.142x2 + 0.028x + -0.456 = 0 Reorder the terms: -0.456 + 0.028x + 0.142x2 = 0 Solving -0.456 + 0.028x + 0.142x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.142 the coefficient of the squared term: Divide each side by '0.142'. -3.211267606 + 0.1971830986x + x2 = 0 Move the constant term to the right: Add '3.211267606' to each side of the equation. -3.211267606 + 0.1971830986x + 3.211267606 + x2 = 0 + 3.211267606 Reorder the terms: -3.211267606 + 3.211267606 + 0.1971830986x + x2 = 0 + 3.211267606 Combine like terms: -3.211267606 + 3.211267606 = 0.000000000 0.000000000 + 0.1971830986x + x2 = 0 + 3.211267606 0.1971830986x + x2 = 0 + 3.211267606 Combine like terms: 0 + 3.211267606 = 3.211267606 0.1971830986x + x2 = 3.211267606 The x term is 0.1971830986x. Take half its coefficient (0.0985915493). Square it (0.009720293593) and add it to both sides. Add '0.009720293593' to each side of the equation. 0.1971830986x + 0.009720293593 + x2 = 3.211267606 + 0.009720293593 Reorder the terms: 0.009720293593 + 0.1971830986x + x2 = 3.211267606 + 0.009720293593 Combine like terms: 3.211267606 + 0.009720293593 = 3.220987899593 0.009720293593 + 0.1971830986x + x2 = 3.220987899593 Factor a perfect square on the left side: (x + 0.0985915493)(x + 0.0985915493) = 3.220987899593 Calculate the square root of the right side: 1.794711091 Break this problem into two subproblems by setting (x + 0.0985915493) equal to 1.794711091 and -1.794711091.Subproblem 1
x + 0.0985915493 = 1.794711091 Simplifying x + 0.0985915493 = 1.794711091 Reorder the terms: 0.0985915493 + x = 1.794711091 Solving 0.0985915493 + x = 1.794711091 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0985915493' to each side of the equation. 0.0985915493 + -0.0985915493 + x = 1.794711091 + -0.0985915493 Combine like terms: 0.0985915493 + -0.0985915493 = 0.0000000000 0.0000000000 + x = 1.794711091 + -0.0985915493 x = 1.794711091 + -0.0985915493 Combine like terms: 1.794711091 + -0.0985915493 = 1.6961195417 x = 1.6961195417 Simplifying x = 1.6961195417Subproblem 2
x + 0.0985915493 = -1.794711091 Simplifying x + 0.0985915493 = -1.794711091 Reorder the terms: 0.0985915493 + x = -1.794711091 Solving 0.0985915493 + x = -1.794711091 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0985915493' to each side of the equation. 0.0985915493 + -0.0985915493 + x = -1.794711091 + -0.0985915493 Combine like terms: 0.0985915493 + -0.0985915493 = 0.0000000000 0.0000000000 + x = -1.794711091 + -0.0985915493 x = -1.794711091 + -0.0985915493 Combine like terms: -1.794711091 + -0.0985915493 = -1.8933026403 x = -1.8933026403 Simplifying x = -1.8933026403Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.6961195417, -1.8933026403}
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