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Simplifying 500 = 45x + 35x(12 + -1x) 500 = 45x + (12 * 35x + -1x * 35x) 500 = 45x + (420x + -35x2) Combine like terms: 45x + 420x = 465x 500 = 465x + -35x2 Solving 500 = 465x + -35x2 Solving for variable 'x'. Reorder the terms: 500 + -465x + 35x2 = 465x + -465x + -35x2 + 35x2 Combine like terms: 465x + -465x = 0 500 + -465x + 35x2 = 0 + -35x2 + 35x2 500 + -465x + 35x2 = -35x2 + 35x2 Combine like terms: -35x2 + 35x2 = 0 500 + -465x + 35x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(100 + -93x + 7x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(100 + -93x + 7x2)' equal to zero and attempt to solve: Simplifying 100 + -93x + 7x2 = 0 Solving 100 + -93x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 14.28571429 + -13.28571429x + x2 = 0 Move the constant term to the right: Add '-14.28571429' to each side of the equation. 14.28571429 + -13.28571429x + -14.28571429 + x2 = 0 + -14.28571429 Reorder the terms: 14.28571429 + -14.28571429 + -13.28571429x + x2 = 0 + -14.28571429 Combine like terms: 14.28571429 + -14.28571429 = 0.00000000 0.00000000 + -13.28571429x + x2 = 0 + -14.28571429 -13.28571429x + x2 = 0 + -14.28571429 Combine like terms: 0 + -14.28571429 = -14.28571429 -13.28571429x + x2 = -14.28571429 The x term is -13.28571429x. Take half its coefficient (-6.642857145). Square it (44.12755105) and add it to both sides. Add '44.12755105' to each side of the equation. -13.28571429x + 44.12755105 + x2 = -14.28571429 + 44.12755105 Reorder the terms: 44.12755105 + -13.28571429x + x2 = -14.28571429 + 44.12755105 Combine like terms: -14.28571429 + 44.12755105 = 29.84183676 44.12755105 + -13.28571429x + x2 = 29.84183676 Factor a perfect square on the left side: (x + -6.642857145)(x + -6.642857145) = 29.84183676 Calculate the square root of the right side: 5.462768232 Break this problem into two subproblems by setting (x + -6.642857145) equal to 5.462768232 and -5.462768232.Subproblem 1
x + -6.642857145 = 5.462768232 Simplifying x + -6.642857145 = 5.462768232 Reorder the terms: -6.642857145 + x = 5.462768232 Solving -6.642857145 + x = 5.462768232 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.642857145' to each side of the equation. -6.642857145 + 6.642857145 + x = 5.462768232 + 6.642857145 Combine like terms: -6.642857145 + 6.642857145 = 0.000000000 0.000000000 + x = 5.462768232 + 6.642857145 x = 5.462768232 + 6.642857145 Combine like terms: 5.462768232 + 6.642857145 = 12.105625377 x = 12.105625377 Simplifying x = 12.105625377Subproblem 2
x + -6.642857145 = -5.462768232 Simplifying x + -6.642857145 = -5.462768232 Reorder the terms: -6.642857145 + x = -5.462768232 Solving -6.642857145 + x = -5.462768232 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.642857145' to each side of the equation. -6.642857145 + 6.642857145 + x = -5.462768232 + 6.642857145 Combine like terms: -6.642857145 + 6.642857145 = 0.000000000 0.000000000 + x = -5.462768232 + 6.642857145 x = -5.462768232 + 6.642857145 Combine like terms: -5.462768232 + 6.642857145 = 1.180088913 x = 1.180088913 Simplifying x = 1.180088913Solution
The solution to the problem is based on the solutions from the subproblems. x = {12.105625377, 1.180088913}Solution
x = {12.105625377, 1.180088913}
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