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Simplifying 2(X + 2) + 3X = 2X(X + 1) + 1 Reorder the terms: 2(2 + X) + 3X = 2X(X + 1) + 1 (2 * 2 + X * 2) + 3X = 2X(X + 1) + 1 (4 + 2X) + 3X = 2X(X + 1) + 1 Combine like terms: 2X + 3X = 5X 4 + 5X = 2X(X + 1) + 1 Reorder the terms: 4 + 5X = 2X(1 + X) + 1 4 + 5X = (1 * 2X + X * 2X) + 1 4 + 5X = (2X + 2X2) + 1 Reorder the terms: 4 + 5X = 1 + 2X + 2X2 Solving 4 + 5X = 1 + 2X + 2X2 Solving for variable 'X'. Reorder the terms: 4 + -1 + 5X + -2X + -2X2 = 1 + 2X + 2X2 + -1 + -2X + -2X2 Combine like terms: 4 + -1 = 3 3 + 5X + -2X + -2X2 = 1 + 2X + 2X2 + -1 + -2X + -2X2 Combine like terms: 5X + -2X = 3X 3 + 3X + -2X2 = 1 + 2X + 2X2 + -1 + -2X + -2X2 Reorder the terms: 3 + 3X + -2X2 = 1 + -1 + 2X + -2X + 2X2 + -2X2 Combine like terms: 1 + -1 = 0 3 + 3X + -2X2 = 0 + 2X + -2X + 2X2 + -2X2 3 + 3X + -2X2 = 2X + -2X + 2X2 + -2X2 Combine like terms: 2X + -2X = 0 3 + 3X + -2X2 = 0 + 2X2 + -2X2 3 + 3X + -2X2 = 2X2 + -2X2 Combine like terms: 2X2 + -2X2 = 0 3 + 3X + -2X2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1.5 + -1.5X + X2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -1.5X + 1.5 + X2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -1.5X + X2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -1.5X + X2 = 0 + 1.5 -1.5X + X2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -1.5X + X2 = 1.5 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 = 1.5 + 0.5625 Reorder the terms: 0.5625 + -1.5X + X2 = 1.5 + 0.5625 Combine like terms: 1.5 + 0.5625 = 2.0625 0.5625 + -1.5X + X2 = 2.0625 Factor a perfect square on the left side: (X + -0.75)(X + -0.75) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (X + -0.75) equal to 1.436140662 and -1.436140662.Subproblem 1
X + -0.75 = 1.436140662 Simplifying X + -0.75 = 1.436140662 Reorder the terms: -0.75 + X = 1.436140662 Solving -0.75 + X = 1.436140662 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 = 1.436140662 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + X = 1.436140662 + 0.75 X = 1.436140662 + 0.75 Combine like terms: 1.436140662 + 0.75 = 2.186140662 X = 2.186140662 Simplifying X = 2.186140662Subproblem 2
X + -0.75 = -1.436140662 Simplifying X + -0.75 = -1.436140662 Reorder the terms: -0.75 + X = -1.436140662 Solving -0.75 + X = -1.436140662 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 = -1.436140662 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + X = -1.436140662 + 0.75 X = -1.436140662 + 0.75 Combine like terms: -1.436140662 + 0.75 = -0.686140662 X = -0.686140662 Simplifying X = -0.686140662Solution
The solution to the problem is based on the solutions from the subproblems. X = {2.186140662, -0.686140662}
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