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Simplifying -23x2 + 54x + 117 = 0 Reorder the terms: 117 + 54x + -23x2 = 0 Solving 117 + 54x + -23x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -23 the coefficient of the squared term: Divide each side by '-23'. -5.086956522 + -2.347826087x + x2 = 0 Move the constant term to the right: Add '5.086956522' to each side of the equation. -5.086956522 + -2.347826087x + 5.086956522 + x2 = 0 + 5.086956522 Reorder the terms: -5.086956522 + 5.086956522 + -2.347826087x + x2 = 0 + 5.086956522 Combine like terms: -5.086956522 + 5.086956522 = 0.000000000 0.000000000 + -2.347826087x + x2 = 0 + 5.086956522 -2.347826087x + x2 = 0 + 5.086956522 Combine like terms: 0 + 5.086956522 = 5.086956522 -2.347826087x + x2 = 5.086956522 The x term is -2.347826087x. Take half its coefficient (-1.173913044). Square it (1.378071835) and add it to both sides. Add '1.378071835' to each side of the equation. -2.347826087x + 1.378071835 + x2 = 5.086956522 + 1.378071835 Reorder the terms: 1.378071835 + -2.347826087x + x2 = 5.086956522 + 1.378071835 Combine like terms: 5.086956522 + 1.378071835 = 6.465028357 1.378071835 + -2.347826087x + x2 = 6.465028357 Factor a perfect square on the left side: (x + -1.173913044)(x + -1.173913044) = 6.465028357 Calculate the square root of the right side: 2.542642003 Break this problem into two subproblems by setting (x + -1.173913044) equal to 2.542642003 and -2.542642003.Subproblem 1
x + -1.173913044 = 2.542642003 Simplifying x + -1.173913044 = 2.542642003 Reorder the terms: -1.173913044 + x = 2.542642003 Solving -1.173913044 + x = 2.542642003 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.173913044' to each side of the equation. -1.173913044 + 1.173913044 + x = 2.542642003 + 1.173913044 Combine like terms: -1.173913044 + 1.173913044 = 0.000000000 0.000000000 + x = 2.542642003 + 1.173913044 x = 2.542642003 + 1.173913044 Combine like terms: 2.542642003 + 1.173913044 = 3.716555047 x = 3.716555047 Simplifying x = 3.716555047Subproblem 2
x + -1.173913044 = -2.542642003 Simplifying x + -1.173913044 = -2.542642003 Reorder the terms: -1.173913044 + x = -2.542642003 Solving -1.173913044 + x = -2.542642003 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.173913044' to each side of the equation. -1.173913044 + 1.173913044 + x = -2.542642003 + 1.173913044 Combine like terms: -1.173913044 + 1.173913044 = 0.000000000 0.000000000 + x = -2.542642003 + 1.173913044 x = -2.542642003 + 1.173913044 Combine like terms: -2.542642003 + 1.173913044 = -1.368728959 x = -1.368728959 Simplifying x = -1.368728959Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.716555047, -1.368728959}
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