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Simplifying 6 = -6X2 + -9X + 9 Reorder the terms: 6 = 9 + -9X + -6X2 Solving 6 = 9 + -9X + -6X2 Solving for variable 'X'. Combine like terms: 6 + -9 = -3 -3 + 9X + 6X2 = 9 + -9X + -6X2 + -9 + 9X + 6X2 Reorder the terms: -3 + 9X + 6X2 = 9 + -9 + -9X + 9X + -6X2 + 6X2 Combine like terms: 9 + -9 = 0 -3 + 9X + 6X2 = 0 + -9X + 9X + -6X2 + 6X2 -3 + 9X + 6X2 = -9X + 9X + -6X2 + 6X2 Combine like terms: -9X + 9X = 0 -3 + 9X + 6X2 = 0 + -6X2 + 6X2 -3 + 9X + 6X2 = -6X2 + 6X2 Combine like terms: -6X2 + 6X2 = 0 -3 + 9X + 6X2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + 3X + 2X2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + 3X + 2X2)' equal to zero and attempt to solve: Simplifying -1 + 3X + 2X2 = 0 Solving -1 + 3X + 2X2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + 1.5X + X2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 1.5X + 0.5 + X2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 1.5X + X2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 1.5X + X2 = 0 + 0.5 1.5X + X2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 1.5X + X2 = 0.5 The X term is 1.5X. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5X + 0.5625 + X2 = 0.5 + 0.5625 Reorder the terms: 0.5625 + 1.5X + X2 = 0.5 + 0.5625 Combine like terms: 0.5 + 0.5625 = 1.0625 0.5625 + 1.5X + X2 = 1.0625 Factor a perfect square on the left side: (X + 0.75)(X + 0.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (X + 0.75) equal to 1.030776406 and -1.030776406.Subproblem 1
X + 0.75 = 1.030776406 Simplifying X + 0.75 = 1.030776406 Reorder the terms: 0.75 + X = 1.030776406 Solving 0.75 + X = 1.030776406 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + X = 1.030776406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + X = 1.030776406 + -0.75 X = 1.030776406 + -0.75 Combine like terms: 1.030776406 + -0.75 = 0.280776406 X = 0.280776406 Simplifying X = 0.280776406Subproblem 2
X + 0.75 = -1.030776406 Simplifying X + 0.75 = -1.030776406 Reorder the terms: 0.75 + X = -1.030776406 Solving 0.75 + X = -1.030776406 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + X = -1.030776406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + X = -1.030776406 + -0.75 X = -1.030776406 + -0.75 Combine like terms: -1.030776406 + -0.75 = -1.780776406 X = -1.780776406 Simplifying X = -1.780776406Solution
The solution to the problem is based on the solutions from the subproblems. X = {0.280776406, -1.780776406}Solution
X = {0.280776406, -1.780776406}
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