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Simplifying (2X + -3)(2X + -3) = 23 Reorder the terms: (-3 + 2X)(2X + -3) = 23 Reorder the terms: (-3 + 2X)(-3 + 2X) = 23 Multiply (-3 + 2X) * (-3 + 2X) (-3(-3 + 2X) + 2X * (-3 + 2X)) = 23 ((-3 * -3 + 2X * -3) + 2X * (-3 + 2X)) = 23 ((9 + -6X) + 2X * (-3 + 2X)) = 23 (9 + -6X + (-3 * 2X + 2X * 2X)) = 23 (9 + -6X + (-6X + 4X2)) = 23 Combine like terms: -6X + -6X = -12X (9 + -12X + 4X2) = 23 Solving 9 + -12X + 4X2 = 23 Solving for variable 'X'. Reorder the terms: 9 + -23 + -12X + 4X2 = 23 + -23 Combine like terms: 9 + -23 = -14 -14 + -12X + 4X2 = 23 + -23 Combine like terms: 23 + -23 = 0 -14 + -12X + 4X2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -6X + 2X2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -6X + 2X2)' equal to zero and attempt to solve: Simplifying -7 + -6X + 2X2 = 0 Solving -7 + -6X + 2X2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + -3X + X2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -3X + 3.5 + X2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -3X + X2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -3X + X2 = 0 + 3.5 -3X + X2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -3X + X2 = 3.5 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 = 3.5 + 2.25 Reorder the terms: 2.25 + -3X + X2 = 3.5 + 2.25 Combine like terms: 3.5 + 2.25 = 5.75 2.25 + -3X + X2 = 5.75 Factor a perfect square on the left side: (X + -1.5)(X + -1.5) = 5.75 Calculate the square root of the right side: 2.397915762 Break this problem into two subproblems by setting (X + -1.5) equal to 2.397915762 and -2.397915762.Subproblem 1
X + -1.5 = 2.397915762 Simplifying X + -1.5 = 2.397915762 Reorder the terms: -1.5 + X = 2.397915762 Solving -1.5 + X = 2.397915762 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 = 2.397915762 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + X = 2.397915762 + 1.5 X = 2.397915762 + 1.5 Combine like terms: 2.397915762 + 1.5 = 3.897915762 X = 3.897915762 Simplifying X = 3.897915762Subproblem 2
X + -1.5 = -2.397915762 Simplifying X + -1.5 = -2.397915762 Reorder the terms: -1.5 + X = -2.397915762 Solving -1.5 + X = -2.397915762 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 = -2.397915762 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + X = -2.397915762 + 1.5 X = -2.397915762 + 1.5 Combine like terms: -2.397915762 + 1.5 = -0.897915762 X = -0.897915762 Simplifying X = -0.897915762Solution
The solution to the problem is based on the solutions from the subproblems. X = {3.897915762, -0.897915762}Solution
X = {3.897915762, -0.897915762}
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