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Simplifying 81x2 + -67x + -80 = 0 Reorder the terms: -80 + -67x + 81x2 = 0 Solving -80 + -67x + 81x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 81 the coefficient of the squared term: Divide each side by '81'. -0.987654321 + -0.8271604938x + x2 = 0 Move the constant term to the right: Add '0.987654321' to each side of the equation. -0.987654321 + -0.8271604938x + 0.987654321 + x2 = 0 + 0.987654321 Reorder the terms: -0.987654321 + 0.987654321 + -0.8271604938x + x2 = 0 + 0.987654321 Combine like terms: -0.987654321 + 0.987654321 = 0.000000000 0.000000000 + -0.8271604938x + x2 = 0 + 0.987654321 -0.8271604938x + x2 = 0 + 0.987654321 Combine like terms: 0 + 0.987654321 = 0.987654321 -0.8271604938x + x2 = 0.987654321 The x term is -0.8271604938x. Take half its coefficient (-0.4135802469). Square it (0.1710486206) and add it to both sides. Add '0.1710486206' to each side of the equation. -0.8271604938x + 0.1710486206 + x2 = 0.987654321 + 0.1710486206 Reorder the terms: 0.1710486206 + -0.8271604938x + x2 = 0.987654321 + 0.1710486206 Combine like terms: 0.987654321 + 0.1710486206 = 1.1587029416 0.1710486206 + -0.8271604938x + x2 = 1.1587029416 Factor a perfect square on the left side: (x + -0.4135802469)(x + -0.4135802469) = 1.1587029416 Calculate the square root of the right side: 1.076430649 Break this problem into two subproblems by setting (x + -0.4135802469) equal to 1.076430649 and -1.076430649.Subproblem 1
x + -0.4135802469 = 1.076430649 Simplifying x + -0.4135802469 = 1.076430649 Reorder the terms: -0.4135802469 + x = 1.076430649 Solving -0.4135802469 + x = 1.076430649 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4135802469' to each side of the equation. -0.4135802469 + 0.4135802469 + x = 1.076430649 + 0.4135802469 Combine like terms: -0.4135802469 + 0.4135802469 = 0.0000000000 0.0000000000 + x = 1.076430649 + 0.4135802469 x = 1.076430649 + 0.4135802469 Combine like terms: 1.076430649 + 0.4135802469 = 1.4900108959 x = 1.4900108959 Simplifying x = 1.4900108959Subproblem 2
x + -0.4135802469 = -1.076430649 Simplifying x + -0.4135802469 = -1.076430649 Reorder the terms: -0.4135802469 + x = -1.076430649 Solving -0.4135802469 + x = -1.076430649 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4135802469' to each side of the equation. -0.4135802469 + 0.4135802469 + x = -1.076430649 + 0.4135802469 Combine like terms: -0.4135802469 + 0.4135802469 = 0.0000000000 0.0000000000 + x = -1.076430649 + 0.4135802469 x = -1.076430649 + 0.4135802469 Combine like terms: -1.076430649 + 0.4135802469 = -0.6628504021 x = -0.6628504021 Simplifying x = -0.6628504021Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.4900108959, -0.6628504021}
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