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Simplifying 69x2 + -68x + -67 = 0 Reorder the terms: -67 + -68x + 69x2 = 0 Solving -67 + -68x + 69x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 69 the coefficient of the squared term: Divide each side by '69'. -0.9710144928 + -0.9855072464x + x2 = 0 Move the constant term to the right: Add '0.9710144928' to each side of the equation. -0.9710144928 + -0.9855072464x + 0.9710144928 + x2 = 0 + 0.9710144928 Reorder the terms: -0.9710144928 + 0.9710144928 + -0.9855072464x + x2 = 0 + 0.9710144928 Combine like terms: -0.9710144928 + 0.9710144928 = 0.0000000000 0.0000000000 + -0.9855072464x + x2 = 0 + 0.9710144928 -0.9855072464x + x2 = 0 + 0.9710144928 Combine like terms: 0 + 0.9710144928 = 0.9710144928 -0.9855072464x + x2 = 0.9710144928 The x term is -0.9855072464x. Take half its coefficient (-0.4927536232). Square it (0.2428061332) and add it to both sides. Add '0.2428061332' to each side of the equation. -0.9855072464x + 0.2428061332 + x2 = 0.9710144928 + 0.2428061332 Reorder the terms: 0.2428061332 + -0.9855072464x + x2 = 0.9710144928 + 0.2428061332 Combine like terms: 0.9710144928 + 0.2428061332 = 1.213820626 0.2428061332 + -0.9855072464x + x2 = 1.213820626 Factor a perfect square on the left side: (x + -0.4927536232)(x + -0.4927536232) = 1.213820626 Calculate the square root of the right side: 1.101735279 Break this problem into two subproblems by setting (x + -0.4927536232) equal to 1.101735279 and -1.101735279.Subproblem 1
x + -0.4927536232 = 1.101735279 Simplifying x + -0.4927536232 = 1.101735279 Reorder the terms: -0.4927536232 + x = 1.101735279 Solving -0.4927536232 + x = 1.101735279 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4927536232' to each side of the equation. -0.4927536232 + 0.4927536232 + x = 1.101735279 + 0.4927536232 Combine like terms: -0.4927536232 + 0.4927536232 = 0.0000000000 0.0000000000 + x = 1.101735279 + 0.4927536232 x = 1.101735279 + 0.4927536232 Combine like terms: 1.101735279 + 0.4927536232 = 1.5944889022 x = 1.5944889022 Simplifying x = 1.5944889022Subproblem 2
x + -0.4927536232 = -1.101735279 Simplifying x + -0.4927536232 = -1.101735279 Reorder the terms: -0.4927536232 + x = -1.101735279 Solving -0.4927536232 + x = -1.101735279 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4927536232' to each side of the equation. -0.4927536232 + 0.4927536232 + x = -1.101735279 + 0.4927536232 Combine like terms: -0.4927536232 + 0.4927536232 = 0.0000000000 0.0000000000 + x = -1.101735279 + 0.4927536232 x = -1.101735279 + 0.4927536232 Combine like terms: -1.101735279 + 0.4927536232 = -0.6089816558 x = -0.6089816558 Simplifying x = -0.6089816558Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5944889022, -0.6089816558}
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