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Simplifying -416.72x2 + 4416.7x + 8500 = 0 Reorder the terms: 8500 + 4416.7x + -416.72x2 = 0 Solving 8500 + 4416.7x + -416.72x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -416.72 the coefficient of the squared term: Divide each side by '-416.72'. -20.39738913 + -10.59872336x + x2 = 0 Move the constant term to the right: Add '20.39738913' to each side of the equation. -20.39738913 + -10.59872336x + 20.39738913 + x2 = 0 + 20.39738913 Reorder the terms: -20.39738913 + 20.39738913 + -10.59872336x + x2 = 0 + 20.39738913 Combine like terms: -20.39738913 + 20.39738913 = 0.00000000 0.00000000 + -10.59872336x + x2 = 0 + 20.39738913 -10.59872336x + x2 = 0 + 20.39738913 Combine like terms: 0 + 20.39738913 = 20.39738913 -10.59872336x + x2 = 20.39738913 The x term is -10.59872336x. Take half its coefficient (-5.29936168). Square it (28.08323422) and add it to both sides. Add '28.08323422' to each side of the equation. -10.59872336x + 28.08323422 + x2 = 20.39738913 + 28.08323422 Reorder the terms: 28.08323422 + -10.59872336x + x2 = 20.39738913 + 28.08323422 Combine like terms: 20.39738913 + 28.08323422 = 48.48062335 28.08323422 + -10.59872336x + x2 = 48.48062335 Factor a perfect square on the left side: (x + -5.29936168)(x + -5.29936168) = 48.48062335 Calculate the square root of the right side: 6.962802837 Break this problem into two subproblems by setting (x + -5.29936168) equal to 6.962802837 and -6.962802837.Subproblem 1
x + -5.29936168 = 6.962802837 Simplifying x + -5.29936168 = 6.962802837 Reorder the terms: -5.29936168 + x = 6.962802837 Solving -5.29936168 + x = 6.962802837 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.29936168' to each side of the equation. -5.29936168 + 5.29936168 + x = 6.962802837 + 5.29936168 Combine like terms: -5.29936168 + 5.29936168 = 0.00000000 0.00000000 + x = 6.962802837 + 5.29936168 x = 6.962802837 + 5.29936168 Combine like terms: 6.962802837 + 5.29936168 = 12.262164517 x = 12.262164517 Simplifying x = 12.262164517Subproblem 2
x + -5.29936168 = -6.962802837 Simplifying x + -5.29936168 = -6.962802837 Reorder the terms: -5.29936168 + x = -6.962802837 Solving -5.29936168 + x = -6.962802837 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.29936168' to each side of the equation. -5.29936168 + 5.29936168 + x = -6.962802837 + 5.29936168 Combine like terms: -5.29936168 + 5.29936168 = 0.00000000 0.00000000 + x = -6.962802837 + 5.29936168 x = -6.962802837 + 5.29936168 Combine like terms: -6.962802837 + 5.29936168 = -1.663441157 x = -1.663441157 Simplifying x = -1.663441157Solution
The solution to the problem is based on the solutions from the subproblems. x = {12.262164517, -1.663441157}
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