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Simplifying x(x + 1) + (x + 2) = 51 Reorder the terms: x(1 + x) + (x + 2) = 51 (1 * x + x * x) + (x + 2) = 51 (1x + x2) + (x + 2) = 51 Reorder the terms: 1x + x2 + (2 + x) = 51 Remove parenthesis around (2 + x) 1x + x2 + 2 + x = 51 Reorder the terms: 2 + 1x + x + x2 = 51 Combine like terms: 1x + x = 2x 2 + 2x + x2 = 51 Solving 2 + 2x + x2 = 51 Solving for variable 'x'. Reorder the terms: 2 + -51 + 2x + x2 = 51 + -51 Combine like terms: 2 + -51 = -49 -49 + 2x + x2 = 51 + -51 Combine like terms: 51 + -51 = 0 -49 + 2x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '49' to each side of the equation. -49 + 2x + 49 + x2 = 0 + 49 Reorder the terms: -49 + 49 + 2x + x2 = 0 + 49 Combine like terms: -49 + 49 = 0 0 + 2x + x2 = 0 + 49 2x + x2 = 0 + 49 Combine like terms: 0 + 49 = 49 2x + x2 = 49 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 49 + 1 Reorder the terms: 1 + 2x + x2 = 49 + 1 Combine like terms: 49 + 1 = 50 1 + 2x + x2 = 50 Factor a perfect square on the left side: (x + 1)(x + 1) = 50 Calculate the square root of the right side: 7.071067812 Break this problem into two subproblems by setting (x + 1) equal to 7.071067812 and -7.071067812.Subproblem 1
x + 1 = 7.071067812 Simplifying x + 1 = 7.071067812 Reorder the terms: 1 + x = 7.071067812 Solving 1 + x = 7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 7.071067812 + -1 Combine like terms: 1 + -1 = 0 0 + x = 7.071067812 + -1 x = 7.071067812 + -1 Combine like terms: 7.071067812 + -1 = 6.071067812 x = 6.071067812 Simplifying x = 6.071067812Subproblem 2
x + 1 = -7.071067812 Simplifying x + 1 = -7.071067812 Reorder the terms: 1 + x = -7.071067812 Solving 1 + x = -7.071067812 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -7.071067812 + -1 Combine like terms: 1 + -1 = 0 0 + x = -7.071067812 + -1 x = -7.071067812 + -1 Combine like terms: -7.071067812 + -1 = -8.071067812 x = -8.071067812 Simplifying x = -8.071067812Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.071067812, -8.071067812}
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