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Simplifying 23x2 + -54x + 12 = 45 Reorder the terms: 12 + -54x + 23x2 = 45 Solving 12 + -54x + 23x2 = 45 Solving for variable 'x'. Reorder the terms: 12 + -45 + -54x + 23x2 = 45 + -45 Combine like terms: 12 + -45 = -33 -33 + -54x + 23x2 = 45 + -45 Combine like terms: 45 + -45 = 0 -33 + -54x + 23x2 = 0 Begin completing the square. Divide all terms by 23 the coefficient of the squared term: Divide each side by '23'. -1.434782609 + -2.347826087x + x2 = 0 Move the constant term to the right: Add '1.434782609' to each side of the equation. -1.434782609 + -2.347826087x + 1.434782609 + x2 = 0 + 1.434782609 Reorder the terms: -1.434782609 + 1.434782609 + -2.347826087x + x2 = 0 + 1.434782609 Combine like terms: -1.434782609 + 1.434782609 = 0.000000000 0.000000000 + -2.347826087x + x2 = 0 + 1.434782609 -2.347826087x + x2 = 0 + 1.434782609 Combine like terms: 0 + 1.434782609 = 1.434782609 -2.347826087x + x2 = 1.434782609 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 = 1.434782609 + 1.378071835 Reorder the terms: 1.378071835 + -2.347826087x + x2 = 1.434782609 + 1.378071835 Combine like terms: 1.434782609 + 1.378071835 = 2.812854444 1.378071835 + -2.347826087x + x2 = 2.812854444 Factor a perfect square on the left side: (x + -1.173913044)(x + -1.173913044) = 2.812854444 Calculate the square root of the right side: 1.677156655 Break this problem into two subproblems by setting (x + -1.173913044) equal to 1.677156655 and -1.677156655.Subproblem 1
x + -1.173913044 = 1.677156655 Simplifying x + -1.173913044 = 1.677156655 Reorder the terms: -1.173913044 + x = 1.677156655 Solving -1.173913044 + x = 1.677156655 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 = 1.677156655 + 1.173913044 Combine like terms: -1.173913044 + 1.173913044 = 0.000000000 0.000000000 + x = 1.677156655 + 1.173913044 x = 1.677156655 + 1.173913044 Combine like terms: 1.677156655 + 1.173913044 = 2.851069699 x = 2.851069699 Simplifying x = 2.851069699Subproblem 2
x + -1.173913044 = -1.677156655 Simplifying x + -1.173913044 = -1.677156655 Reorder the terms: -1.173913044 + x = -1.677156655 Solving -1.173913044 + x = -1.677156655 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 = -1.677156655 + 1.173913044 Combine like terms: -1.173913044 + 1.173913044 = 0.000000000 0.000000000 + x = -1.677156655 + 1.173913044 x = -1.677156655 + 1.173913044 Combine like terms: -1.677156655 + 1.173913044 = -0.503243611 x = -0.503243611 Simplifying x = -0.503243611Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.851069699, -0.503243611}
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