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Simplifying -416.72x2 + 4416.7x + 8500 = 12354 Reorder the terms: 8500 + 4416.7x + -416.72x2 = 12354 Solving 8500 + 4416.7x + -416.72x2 = 12354 Solving for variable 'x'. Reorder the terms: 8500 + -12354 + 4416.7x + -416.72x2 = 12354 + -12354 Combine like terms: 8500 + -12354 = -3854 -3854 + 4416.7x + -416.72x2 = 12354 + -12354 Combine like terms: 12354 + -12354 = 0 -3854 + 4416.7x + -416.72x2 = 0 Begin completing the square. Divide all terms by -416.72 the coefficient of the squared term: Divide each side by '-416.72'. 9.248416203 + -10.59872336x + x2 = 0 Move the constant term to the right: Add '-9.248416203' to each side of the equation. 9.248416203 + -10.59872336x + -9.248416203 + x2 = 0 + -9.248416203 Reorder the terms: 9.248416203 + -9.248416203 + -10.59872336x + x2 = 0 + -9.248416203 Combine like terms: 9.248416203 + -9.248416203 = 0.000000000 0.000000000 + -10.59872336x + x2 = 0 + -9.248416203 -10.59872336x + x2 = 0 + -9.248416203 Combine like terms: 0 + -9.248416203 = -9.248416203 -10.59872336x + x2 = -9.248416203 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 = -9.248416203 + 28.08323422 Reorder the terms: 28.08323422 + -10.59872336x + x2 = -9.248416203 + 28.08323422 Combine like terms: -9.248416203 + 28.08323422 = 18.834818017 28.08323422 + -10.59872336x + x2 = 18.834818017 Factor a perfect square on the left side: (x + -5.29936168)(x + -5.29936168) = 18.834818017 Calculate the square root of the right side: 4.339909909 Break this problem into two subproblems by setting (x + -5.29936168) equal to 4.339909909 and -4.339909909.Subproblem 1
x + -5.29936168 = 4.339909909 Simplifying x + -5.29936168 = 4.339909909 Reorder the terms: -5.29936168 + x = 4.339909909 Solving -5.29936168 + x = 4.339909909 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 = 4.339909909 + 5.29936168 Combine like terms: -5.29936168 + 5.29936168 = 0.00000000 0.00000000 + x = 4.339909909 + 5.29936168 x = 4.339909909 + 5.29936168 Combine like terms: 4.339909909 + 5.29936168 = 9.639271589 x = 9.639271589 Simplifying x = 9.639271589Subproblem 2
x + -5.29936168 = -4.339909909 Simplifying x + -5.29936168 = -4.339909909 Reorder the terms: -5.29936168 + x = -4.339909909 Solving -5.29936168 + x = -4.339909909 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 = -4.339909909 + 5.29936168 Combine like terms: -5.29936168 + 5.29936168 = 0.00000000 0.00000000 + x = -4.339909909 + 5.29936168 x = -4.339909909 + 5.29936168 Combine like terms: -4.339909909 + 5.29936168 = 0.959451771 x = 0.959451771 Simplifying x = 0.959451771Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.639271589, 0.959451771}
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