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Simplifying 2X(X + 1) = 11 Reorder the terms: 2X(1 + X) = 11 (1 * 2X + X * 2X) = 11 (2X + 2X2) = 11 Solving 2X + 2X2 = 11 Solving for variable 'X'. Reorder the terms: -11 + 2X + 2X2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + 2X + 2X2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -5.5 + X + X2 = 0 Move the constant term to the right: Add '5.5' to each side of the equation. -5.5 + X + 5.5 + X2 = 0 + 5.5 Reorder the terms: -5.5 + 5.5 + X + X2 = 0 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + X + X2 = 0 + 5.5 X + X2 = 0 + 5.5 Combine like terms: 0 + 5.5 = 5.5 X + X2 = 5.5 The X term is X. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + X2 = 5.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + X2 = 5.5 + 0.25 Combine like terms: 5.5 + 0.25 = 5.75 1.25 + X2 = 5.75 Factor a perfect square on the left side: (X + 0.5)(X + 0.5) = 5.75 Calculate the square root of the right side: 2.397915762 Break this problem into two subproblems by setting (X + 0.5) equal to 2.397915762 and -2.397915762.Subproblem 1
X + 0.5 = 2.397915762 Simplifying X + 0.5 = 2.397915762 Reorder the terms: 0.5 + X = 2.397915762 Solving 0.5 + X = 2.397915762 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = 2.397915762 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = 2.397915762 + -0.5 X = 2.397915762 + -0.5 Combine like terms: 2.397915762 + -0.5 = 1.897915762 X = 1.897915762 Simplifying X = 1.897915762Subproblem 2
X + 0.5 = -2.397915762 Simplifying X + 0.5 = -2.397915762 Reorder the terms: 0.5 + X = -2.397915762 Solving 0.5 + X = -2.397915762 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + X = -2.397915762 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + X = -2.397915762 + -0.5 X = -2.397915762 + -0.5 Combine like terms: -2.397915762 + -0.5 = -2.897915762 X = -2.897915762 Simplifying X = -2.897915762Solution
The solution to the problem is based on the solutions from the subproblems. X = {1.897915762, -2.897915762}
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