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Simplifying (2x + 1)(x) = 56 Reorder the terms: (1 + 2x)(x) = 56 Reorder the terms for easier multiplication: x(1 + 2x) = 56 (1 * x + 2x * x) = 56 (1x + 2x2) = 56 Solving 1x + 2x2 = 56 Solving for variable 'x'. Reorder the terms: -56 + 1x + 2x2 = 56 + -56 Combine like terms: 56 + -56 = 0 -56 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -28 + 0.5x + x2 = 0 Move the constant term to the right: Add '28' to each side of the equation. -28 + 0.5x + 28 + x2 = 0 + 28 Reorder the terms: -28 + 28 + 0.5x + x2 = 0 + 28 Combine like terms: -28 + 28 = 0 0 + 0.5x + x2 = 0 + 28 0.5x + x2 = 0 + 28 Combine like terms: 0 + 28 = 28 0.5x + x2 = 28 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 28 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 28 + 0.0625 Combine like terms: 28 + 0.0625 = 28.0625 0.0625 + 0.5x + x2 = 28.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 28.0625 Calculate the square root of the right side: 5.297405025 Break this problem into two subproblems by setting (x + 0.25) equal to 5.297405025 and -5.297405025.Subproblem 1
x + 0.25 = 5.297405025 Simplifying x + 0.25 = 5.297405025 Reorder the terms: 0.25 + x = 5.297405025 Solving 0.25 + x = 5.297405025 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 5.297405025 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.297405025 + -0.25 x = 5.297405025 + -0.25 Combine like terms: 5.297405025 + -0.25 = 5.047405025 x = 5.047405025 Simplifying x = 5.047405025Subproblem 2
x + 0.25 = -5.297405025 Simplifying x + 0.25 = -5.297405025 Reorder the terms: 0.25 + x = -5.297405025 Solving 0.25 + x = -5.297405025 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -5.297405025 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.297405025 + -0.25 x = -5.297405025 + -0.25 Combine like terms: -5.297405025 + -0.25 = -5.547405025 x = -5.547405025 Simplifying x = -5.547405025Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.047405025, -5.547405025}
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