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Simplifying 10 = 15(5 + -1x) * x Reorder the terms for easier multiplication: 10 = 15x(5 + -1x) 10 = (5 * 15x + -1x * 15x) 10 = (75x + -15x2) Solving 10 = 75x + -15x2 Solving for variable 'x'. Reorder the terms: 10 + -75x + 15x2 = 75x + -75x + -15x2 + 15x2 Combine like terms: 75x + -75x = 0 10 + -75x + 15x2 = 0 + -15x2 + 15x2 10 + -75x + 15x2 = -15x2 + 15x2 Combine like terms: -15x2 + 15x2 = 0 10 + -75x + 15x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(2 + -15x + 3x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(2 + -15x + 3x2)' equal to zero and attempt to solve: Simplifying 2 + -15x + 3x2 = 0 Solving 2 + -15x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + -5x + x2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -5x + -0.6666666667 + x2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -5x + x2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -5x + x2 = 0 + -0.6666666667 -5x + x2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -5x + x2 = -0.6666666667 The x term is -5x. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5x + 6.25 + x2 = -0.6666666667 + 6.25 Reorder the terms: 6.25 + -5x + x2 = -0.6666666667 + 6.25 Combine like terms: -0.6666666667 + 6.25 = 5.5833333333 6.25 + -5x + x2 = 5.5833333333 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 5.5833333333 Calculate the square root of the right side: 2.362907813 Break this problem into two subproblems by setting (x + -2.5) equal to 2.362907813 and -2.362907813.Subproblem 1
x + -2.5 = 2.362907813 Simplifying x + -2.5 = 2.362907813 Reorder the terms: -2.5 + x = 2.362907813 Solving -2.5 + x = 2.362907813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 2.362907813 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.362907813 + 2.5 x = 2.362907813 + 2.5 Combine like terms: 2.362907813 + 2.5 = 4.862907813 x = 4.862907813 Simplifying x = 4.862907813Subproblem 2
x + -2.5 = -2.362907813 Simplifying x + -2.5 = -2.362907813 Reorder the terms: -2.5 + x = -2.362907813 Solving -2.5 + x = -2.362907813 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -2.362907813 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.362907813 + 2.5 x = -2.362907813 + 2.5 Combine like terms: -2.362907813 + 2.5 = 0.137092187 x = 0.137092187 Simplifying x = 0.137092187Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.862907813, 0.137092187}Solution
x = {4.862907813, 0.137092187}
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