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Simplifying 0 = -4x2 + 46x + -114 Reorder the terms: 0 = -114 + 46x + -4x2 Solving 0 = -114 + 46x + -4x2 Solving for variable 'x'. Combine like terms: 0 + 114 = 114 114 + -46x + 4x2 = -114 + 46x + -4x2 + 114 + -46x + 4x2 Reorder the terms: 114 + -46x + 4x2 = -114 + 114 + 46x + -46x + -4x2 + 4x2 Combine like terms: -114 + 114 = 0 114 + -46x + 4x2 = 0 + 46x + -46x + -4x2 + 4x2 114 + -46x + 4x2 = 46x + -46x + -4x2 + 4x2 Combine like terms: 46x + -46x = 0 114 + -46x + 4x2 = 0 + -4x2 + 4x2 114 + -46x + 4x2 = -4x2 + 4x2 Combine like terms: -4x2 + 4x2 = 0 114 + -46x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(57 + -23x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(57 + -23x + 2x2)' equal to zero and attempt to solve: Simplifying 57 + -23x + 2x2 = 0 Solving 57 + -23x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 28.5 + -11.5x + x2 = 0 Move the constant term to the right: Add '-28.5' to each side of the equation. 28.5 + -11.5x + -28.5 + x2 = 0 + -28.5 Reorder the terms: 28.5 + -28.5 + -11.5x + x2 = 0 + -28.5 Combine like terms: 28.5 + -28.5 = 0.0 0.0 + -11.5x + x2 = 0 + -28.5 -11.5x + x2 = 0 + -28.5 Combine like terms: 0 + -28.5 = -28.5 -11.5x + x2 = -28.5 The x term is -11.5x. Take half its coefficient (-5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. -11.5x + 33.0625 + x2 = -28.5 + 33.0625 Reorder the terms: 33.0625 + -11.5x + x2 = -28.5 + 33.0625 Combine like terms: -28.5 + 33.0625 = 4.5625 33.0625 + -11.5x + x2 = 4.5625 Factor a perfect square on the left side: (x + -5.75)(x + -5.75) = 4.5625 Calculate the square root of the right side: 2.136000936 Break this problem into two subproblems by setting (x + -5.75) equal to 2.136000936 and -2.136000936.Subproblem 1
x + -5.75 = 2.136000936 Simplifying x + -5.75 = 2.136000936 Reorder the terms: -5.75 + x = 2.136000936 Solving -5.75 + x = 2.136000936 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.75' to each side of the equation. -5.75 + 5.75 + x = 2.136000936 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + x = 2.136000936 + 5.75 x = 2.136000936 + 5.75 Combine like terms: 2.136000936 + 5.75 = 7.886000936 x = 7.886000936 Simplifying x = 7.886000936Subproblem 2
x + -5.75 = -2.136000936 Simplifying x + -5.75 = -2.136000936 Reorder the terms: -5.75 + x = -2.136000936 Solving -5.75 + x = -2.136000936 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.75' to each side of the equation. -5.75 + 5.75 + x = -2.136000936 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + x = -2.136000936 + 5.75 x = -2.136000936 + 5.75 Combine like terms: -2.136000936 + 5.75 = 3.613999064 x = 3.613999064 Simplifying x = 3.613999064Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.886000936, 3.613999064}Solution
x = {7.886000936, 3.613999064}
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