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Simplifying 27x2 + -7x + -9 = 0 Reorder the terms: -9 + -7x + 27x2 = 0 Solving -9 + -7x + 27x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.3333333333 + -0.2592592593x + x2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -0.2592592593x + 0.3333333333 + x2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -0.2592592593x + x2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -0.2592592593x + x2 = 0 + 0.3333333333 -0.2592592593x + x2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -0.2592592593x + x2 = 0.3333333333 The x term is -0.2592592593x. Take half its coefficient (-0.1296296297). Square it (0.01680384090) and add it to both sides. Add '0.01680384090' to each side of the equation. -0.2592592593x + 0.01680384090 + x2 = 0.3333333333 + 0.01680384090 Reorder the terms: 0.01680384090 + -0.2592592593x + x2 = 0.3333333333 + 0.01680384090 Combine like terms: 0.3333333333 + 0.01680384090 = 0.3501371742 0.01680384090 + -0.2592592593x + x2 = 0.3501371742 Factor a perfect square on the left side: (x + -0.1296296297)(x + -0.1296296297) = 0.3501371742 Calculate the square root of the right side: 0.5917239 Break this problem into two subproblems by setting (x + -0.1296296297) equal to 0.5917239 and -0.5917239.Subproblem 1
x + -0.1296296297 = 0.5917239 Simplifying x + -0.1296296297 = 0.5917239 Reorder the terms: -0.1296296297 + x = 0.5917239 Solving -0.1296296297 + x = 0.5917239 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1296296297' to each side of the equation. -0.1296296297 + 0.1296296297 + x = 0.5917239 + 0.1296296297 Combine like terms: -0.1296296297 + 0.1296296297 = 0.0000000000 0.0000000000 + x = 0.5917239 + 0.1296296297 x = 0.5917239 + 0.1296296297 Combine like terms: 0.5917239 + 0.1296296297 = 0.7213535297 x = 0.7213535297 Simplifying x = 0.7213535297Subproblem 2
x + -0.1296296297 = -0.5917239 Simplifying x + -0.1296296297 = -0.5917239 Reorder the terms: -0.1296296297 + x = -0.5917239 Solving -0.1296296297 + x = -0.5917239 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1296296297' to each side of the equation. -0.1296296297 + 0.1296296297 + x = -0.5917239 + 0.1296296297 Combine like terms: -0.1296296297 + 0.1296296297 = 0.0000000000 0.0000000000 + x = -0.5917239 + 0.1296296297 x = -0.5917239 + 0.1296296297 Combine like terms: -0.5917239 + 0.1296296297 = -0.4620942703 x = -0.4620942703 Simplifying x = -0.4620942703Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7213535297, -0.4620942703}
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