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Simplifying 7 = -416.72x2 + 4416.7x + 8500 Reorder the terms: 7 = 8500 + 4416.7x + -416.72x2 Solving 7 = 8500 + 4416.7x + -416.72x2 Solving for variable 'x'. Combine like terms: 7 + -8500 = -8493 -8493 + -4416.7x + 416.72x2 = 8500 + 4416.7x + -416.72x2 + -8500 + -4416.7x + 416.72x2 Reorder the terms: -8493 + -4416.7x + 416.72x2 = 8500 + -8500 + 4416.7x + -4416.7x + -416.72x2 + 416.72x2 Combine like terms: 8500 + -8500 = 0 -8493 + -4416.7x + 416.72x2 = 0 + 4416.7x + -4416.7x + -416.72x2 + 416.72x2 -8493 + -4416.7x + 416.72x2 = 4416.7x + -4416.7x + -416.72x2 + 416.72x2 Combine like terms: 4416.7x + -4416.7x = 0.0 -8493 + -4416.7x + 416.72x2 = 0.0 + -416.72x2 + 416.72x2 -8493 + -4416.7x + 416.72x2 = -416.72x2 + 416.72x2 Combine like terms: -416.72x2 + 416.72x2 = 0.00 -8493 + -4416.7x + 416.72x2 = 0.00 Begin completing the square. Divide all terms by 416.72 the coefficient of the squared term: Divide each side by '416.72'. -20.38059128 + -10.59872336x + x2 = 0 Move the constant term to the right: Add '20.38059128' to each side of the equation. -20.38059128 + -10.59872336x + 20.38059128 + x2 = 0 + 20.38059128 Reorder the terms: -20.38059128 + 20.38059128 + -10.59872336x + x2 = 0 + 20.38059128 Combine like terms: -20.38059128 + 20.38059128 = 0.00000000 0.00000000 + -10.59872336x + x2 = 0 + 20.38059128 -10.59872336x + x2 = 0 + 20.38059128 Combine like terms: 0 + 20.38059128 = 20.38059128 -10.59872336x + x2 = 20.38059128 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 = 20.38059128 + 28.08323422 Reorder the terms: 28.08323422 + -10.59872336x + x2 = 20.38059128 + 28.08323422 Combine like terms: 20.38059128 + 28.08323422 = 48.4638255 28.08323422 + -10.59872336x + x2 = 48.4638255 Factor a perfect square on the left side: (x + -5.29936168)(x + -5.29936168) = 48.4638255 Calculate the square root of the right side: 6.961596476 Break this problem into two subproblems by setting (x + -5.29936168) equal to 6.961596476 and -6.961596476.Subproblem 1
x + -5.29936168 = 6.961596476 Simplifying x + -5.29936168 = 6.961596476 Reorder the terms: -5.29936168 + x = 6.961596476 Solving -5.29936168 + x = 6.961596476 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 = 6.961596476 + 5.29936168 Combine like terms: -5.29936168 + 5.29936168 = 0.00000000 0.00000000 + x = 6.961596476 + 5.29936168 x = 6.961596476 + 5.29936168 Combine like terms: 6.961596476 + 5.29936168 = 12.260958156 x = 12.260958156 Simplifying x = 12.260958156Subproblem 2
x + -5.29936168 = -6.961596476 Simplifying x + -5.29936168 = -6.961596476 Reorder the terms: -5.29936168 + x = -6.961596476 Solving -5.29936168 + x = -6.961596476 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 = -6.961596476 + 5.29936168 Combine like terms: -5.29936168 + 5.29936168 = 0.00000000 0.00000000 + x = -6.961596476 + 5.29936168 x = -6.961596476 + 5.29936168 Combine like terms: -6.961596476 + 5.29936168 = -1.662234796 x = -1.662234796 Simplifying x = -1.662234796Solution
The solution to the problem is based on the solutions from the subproblems. x = {12.260958156, -1.662234796}
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