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Simplifying -314.011706x2 + 24672.00111525761x + 1 = 0 Reorder the terms: 1 + 24672.00111525761x + -314.011706x2 = 0 Solving 1 + 24672.00111525761x + -314.011706x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -314.011706 the coefficient of the squared term: Divide each side by '-314.011706'. -0.003184594653 + -78.57032284x + x2 = 0 Move the constant term to the right: Add '0.003184594653' to each side of the equation. -0.003184594653 + -78.57032284x + 0.003184594653 + x2 = 0 + 0.003184594653 Reorder the terms: -0.003184594653 + 0.003184594653 + -78.57032284x + x2 = 0 + 0.003184594653 Combine like terms: -0.003184594653 + 0.003184594653 = 0.000000000000 0.000000000000 + -78.57032284x + x2 = 0 + 0.003184594653 -78.57032284x + x2 = 0 + 0.003184594653 Combine like terms: 0 + 0.003184594653 = 0.003184594653 -78.57032284x + x2 = 0.003184594653 The x term is -78.57032284x. Take half its coefficient (-39.28516142). Square it (1543.323908) and add it to both sides. Add '1543.323908' to each side of the equation. -78.57032284x + 1543.323908 + x2 = 0.003184594653 + 1543.323908 Reorder the terms: 1543.323908 + -78.57032284x + x2 = 0.003184594653 + 1543.323908 Combine like terms: 0.003184594653 + 1543.323908 = 1543.327092594653 1543.323908 + -78.57032284x + x2 = 1543.327092594653 Factor a perfect square on the left side: (x + -39.28516142)(x + -39.28516142) = 1543.327092594653 Calculate the square root of the right side: 39.285201954 Break this problem into two subproblems by setting (x + -39.28516142) equal to 39.285201954 and -39.285201954.Subproblem 1
x + -39.28516142 = 39.285201954 Simplifying x + -39.28516142 = 39.285201954 Reorder the terms: -39.28516142 + x = 39.285201954 Solving -39.28516142 + x = 39.285201954 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '39.28516142' to each side of the equation. -39.28516142 + 39.28516142 + x = 39.285201954 + 39.28516142 Combine like terms: -39.28516142 + 39.28516142 = 0.00000000 0.00000000 + x = 39.285201954 + 39.28516142 x = 39.285201954 + 39.28516142 Combine like terms: 39.285201954 + 39.28516142 = 78.570363374 x = 78.570363374 Simplifying x = 78.570363374Subproblem 2
x + -39.28516142 = -39.285201954 Simplifying x + -39.28516142 = -39.285201954 Reorder the terms: -39.28516142 + x = -39.285201954 Solving -39.28516142 + x = -39.285201954 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '39.28516142' to each side of the equation. -39.28516142 + 39.28516142 + x = -39.285201954 + 39.28516142 Combine like terms: -39.28516142 + 39.28516142 = 0.00000000 0.00000000 + x = -39.285201954 + 39.28516142 x = -39.285201954 + 39.28516142 Combine like terms: -39.285201954 + 39.28516142 = -0.000040534 x = -0.000040534 Simplifying x = -0.000040534Solution
The solution to the problem is based on the solutions from the subproblems. x = {78.570363374, -0.000040534}
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