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Simplifying 196x2 + -196x + -98 = 0 Reorder the terms: -98 + -196x + 196x2 = 0 Solving -98 + -196x + 196x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '98'. 98(-1 + -2x + 2x2) = 0 Ignore the factor 98.Subproblem 1
Set the factor '(-1 + -2x + 2x2)' equal to zero and attempt to solve: Simplifying -1 + -2x + 2x2 = 0 Solving -1 + -2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + -1x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1x + x2 = 0 + 0.5 -1x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1x + x2 = 0.5 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = 0.5 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + -1x + x2 = 0.75 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + -0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + -0.5 = 0.866025404 Simplifying x + -0.5 = 0.866025404 Reorder the terms: -0.5 + x = 0.866025404 Solving -0.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.866025404 + 0.5 x = 0.866025404 + 0.5 Combine like terms: 0.866025404 + 0.5 = 1.366025404 x = 1.366025404 Simplifying x = 1.366025404Subproblem 2
x + -0.5 = -0.866025404 Simplifying x + -0.5 = -0.866025404 Reorder the terms: -0.5 + x = -0.866025404 Solving -0.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -0.866025404 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.866025404 + 0.5 x = -0.866025404 + 0.5 Combine like terms: -0.866025404 + 0.5 = -0.366025404 x = -0.366025404 Simplifying x = -0.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.366025404, -0.366025404}Solution
x = {1.366025404, -0.366025404}
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