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Simplifying 5000 = 2500 + 350x + 25x2 Solving 5000 = 2500 + 350x + 25x2 Solving for variable 'x'. Combine like terms: 5000 + -2500 = 2500 2500 + -350x + -25x2 = 2500 + 350x + 25x2 + -2500 + -350x + -25x2 Reorder the terms: 2500 + -350x + -25x2 = 2500 + -2500 + 350x + -350x + 25x2 + -25x2 Combine like terms: 2500 + -2500 = 0 2500 + -350x + -25x2 = 0 + 350x + -350x + 25x2 + -25x2 2500 + -350x + -25x2 = 350x + -350x + 25x2 + -25x2 Combine like terms: 350x + -350x = 0 2500 + -350x + -25x2 = 0 + 25x2 + -25x2 2500 + -350x + -25x2 = 25x2 + -25x2 Combine like terms: 25x2 + -25x2 = 0 2500 + -350x + -25x2 = 0 Factor out the Greatest Common Factor (GCF), '25'. 25(100 + -14x + -1x2) = 0 Ignore the factor 25.Subproblem 1
Set the factor '(100 + -14x + -1x2)' equal to zero and attempt to solve: Simplifying 100 + -14x + -1x2 = 0 Solving 100 + -14x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -100 + 14x + x2 = 0 Move the constant term to the right: Add '100' to each side of the equation. -100 + 14x + 100 + x2 = 0 + 100 Reorder the terms: -100 + 100 + 14x + x2 = 0 + 100 Combine like terms: -100 + 100 = 0 0 + 14x + x2 = 0 + 100 14x + x2 = 0 + 100 Combine like terms: 0 + 100 = 100 14x + x2 = 100 The x term is 14x. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14x + 49 + x2 = 100 + 49 Reorder the terms: 49 + 14x + x2 = 100 + 49 Combine like terms: 100 + 49 = 149 49 + 14x + x2 = 149 Factor a perfect square on the left side: (x + 7)(x + 7) = 149 Calculate the square root of the right side: 12.206555616 Break this problem into two subproblems by setting (x + 7) equal to 12.206555616 and -12.206555616.Subproblem 1
x + 7 = 12.206555616 Simplifying x + 7 = 12.206555616 Reorder the terms: 7 + x = 12.206555616 Solving 7 + x = 12.206555616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + x = 12.206555616 + -7 Combine like terms: 7 + -7 = 0 0 + x = 12.206555616 + -7 x = 12.206555616 + -7 Combine like terms: 12.206555616 + -7 = 5.206555616 x = 5.206555616 Simplifying x = 5.206555616Subproblem 2
x + 7 = -12.206555616 Simplifying x + 7 = -12.206555616 Reorder the terms: 7 + x = -12.206555616 Solving 7 + x = -12.206555616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + x = -12.206555616 + -7 Combine like terms: 7 + -7 = 0 0 + x = -12.206555616 + -7 x = -12.206555616 + -7 Combine like terms: -12.206555616 + -7 = -19.206555616 x = -19.206555616 Simplifying x = -19.206555616Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.206555616, -19.206555616}Solution
x = {5.206555616, -19.206555616}
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