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Simplifying x + 2x(2x + 5) = 100 Reorder the terms: x + 2x(5 + 2x) = 100 x + (5 * 2x + 2x * 2x) = 100 x + (10x + 4x2) = 100 Combine like terms: x + 10x = 11x 11x + 4x2 = 100 Solving 11x + 4x2 = 100 Solving for variable 'x'. Reorder the terms: -100 + 11x + 4x2 = 100 + -100 Combine like terms: 100 + -100 = 0 -100 + 11x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -25 + 2.75x + x2 = 0 Move the constant term to the right: Add '25' to each side of the equation. -25 + 2.75x + 25 + x2 = 0 + 25 Reorder the terms: -25 + 25 + 2.75x + x2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + 2.75x + x2 = 0 + 25 2.75x + x2 = 0 + 25 Combine like terms: 0 + 25 = 25 2.75x + x2 = 25 The x term is 2.75x. Take half its coefficient (1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. 2.75x + 1.890625 + x2 = 25 + 1.890625 Reorder the terms: 1.890625 + 2.75x + x2 = 25 + 1.890625 Combine like terms: 25 + 1.890625 = 26.890625 1.890625 + 2.75x + x2 = 26.890625 Factor a perfect square on the left side: (x + 1.375)(x + 1.375) = 26.890625 Calculate the square root of the right side: 5.185617128 Break this problem into two subproblems by setting (x + 1.375) equal to 5.185617128 and -5.185617128.Subproblem 1
x + 1.375 = 5.185617128 Simplifying x + 1.375 = 5.185617128 Reorder the terms: 1.375 + x = 5.185617128 Solving 1.375 + x = 5.185617128 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = 5.185617128 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = 5.185617128 + -1.375 x = 5.185617128 + -1.375 Combine like terms: 5.185617128 + -1.375 = 3.810617128 x = 3.810617128 Simplifying x = 3.810617128Subproblem 2
x + 1.375 = -5.185617128 Simplifying x + 1.375 = -5.185617128 Reorder the terms: 1.375 + x = -5.185617128 Solving 1.375 + x = -5.185617128 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + x = -5.185617128 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + x = -5.185617128 + -1.375 x = -5.185617128 + -1.375 Combine like terms: -5.185617128 + -1.375 = -6.560617128 x = -6.560617128 Simplifying x = -6.560617128Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.810617128, -6.560617128}
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