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Simplifying 3(2X + -3)(2X + -3) + -5 = 0 Reorder the terms: 3(-3 + 2X)(2X + -3) + -5 = 0 Reorder the terms: 3(-3 + 2X)(-3 + 2X) + -5 = 0 Multiply (-3 + 2X) * (-3 + 2X) 3(-3(-3 + 2X) + 2X * (-3 + 2X)) + -5 = 0 3((-3 * -3 + 2X * -3) + 2X * (-3 + 2X)) + -5 = 0 3((9 + -6X) + 2X * (-3 + 2X)) + -5 = 0 3(9 + -6X + (-3 * 2X + 2X * 2X)) + -5 = 0 3(9 + -6X + (-6X + 4X2)) + -5 = 0 Combine like terms: -6X + -6X = -12X 3(9 + -12X + 4X2) + -5 = 0 (9 * 3 + -12X * 3 + 4X2 * 3) + -5 = 0 (27 + -36X + 12X2) + -5 = 0 Reorder the terms: 27 + -5 + -36X + 12X2 = 0 Combine like terms: 27 + -5 = 22 22 + -36X + 12X2 = 0 Solving 22 + -36X + 12X2 = 0 Solving for variable 'X'. Factor out the Greatest Common Factor (GCF), '2'. 2(11 + -18X + 6X2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(11 + -18X + 6X2)' equal to zero and attempt to solve: Simplifying 11 + -18X + 6X2 = 0 Solving 11 + -18X + 6X2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 1.833333333 + -3X + X2 = 0 Move the constant term to the right: Add '-1.833333333' to each side of the equation. 1.833333333 + -3X + -1.833333333 + X2 = 0 + -1.833333333 Reorder the terms: 1.833333333 + -1.833333333 + -3X + X2 = 0 + -1.833333333 Combine like terms: 1.833333333 + -1.833333333 = 0.000000000 0.000000000 + -3X + X2 = 0 + -1.833333333 -3X + X2 = 0 + -1.833333333 Combine like terms: 0 + -1.833333333 = -1.833333333 -3X + X2 = -1.833333333 The X term is -3X. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3X + 2.25 + X2 = -1.833333333 + 2.25 Reorder the terms: 2.25 + -3X + X2 = -1.833333333 + 2.25 Combine like terms: -1.833333333 + 2.25 = 0.416666667 2.25 + -3X + X2 = 0.416666667 Factor a perfect square on the left side: (X + -1.5)(X + -1.5) = 0.416666667 Calculate the square root of the right side: 0.645497225 Break this problem into two subproblems by setting (X + -1.5) equal to 0.645497225 and -0.645497225.Subproblem 1
X + -1.5 = 0.645497225 Simplifying X + -1.5 = 0.645497225 Reorder the terms: -1.5 + X = 0.645497225 Solving -1.5 + X = 0.645497225 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + X = 0.645497225 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + X = 0.645497225 + 1.5 X = 0.645497225 + 1.5 Combine like terms: 0.645497225 + 1.5 = 2.145497225 X = 2.145497225 Simplifying X = 2.145497225Subproblem 2
X + -1.5 = -0.645497225 Simplifying X + -1.5 = -0.645497225 Reorder the terms: -1.5 + X = -0.645497225 Solving -1.5 + X = -0.645497225 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + X = -0.645497225 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + X = -0.645497225 + 1.5 X = -0.645497225 + 1.5 Combine like terms: -0.645497225 + 1.5 = 0.854502775 X = 0.854502775 Simplifying X = 0.854502775Solution
The solution to the problem is based on the solutions from the subproblems. X = {2.145497225, 0.854502775}Solution
X = {2.145497225, 0.854502775}
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