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Simplifying (2x + 1)(x) = 648 Reorder the terms: (1 + 2x)(x) = 648 Reorder the terms for easier multiplication: x(1 + 2x) = 648 (1 * x + 2x * x) = 648 (1x + 2x2) = 648 Solving 1x + 2x2 = 648 Solving for variable 'x'. Reorder the terms: -648 + 1x + 2x2 = 648 + -648 Combine like terms: 648 + -648 = 0 -648 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -324 + 0.5x + x2 = 0 Move the constant term to the right: Add '324' to each side of the equation. -324 + 0.5x + 324 + x2 = 0 + 324 Reorder the terms: -324 + 324 + 0.5x + x2 = 0 + 324 Combine like terms: -324 + 324 = 0 0 + 0.5x + x2 = 0 + 324 0.5x + x2 = 0 + 324 Combine like terms: 0 + 324 = 324 0.5x + x2 = 324 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 = 324 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 324 + 0.0625 Combine like terms: 324 + 0.0625 = 324.0625 0.0625 + 0.5x + x2 = 324.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 324.0625 Calculate the square root of the right side: 18.001736027 Break this problem into two subproblems by setting (x + 0.25) equal to 18.001736027 and -18.001736027.Subproblem 1
x + 0.25 = 18.001736027 Simplifying x + 0.25 = 18.001736027 Reorder the terms: 0.25 + x = 18.001736027 Solving 0.25 + x = 18.001736027 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 = 18.001736027 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 18.001736027 + -0.25 x = 18.001736027 + -0.25 Combine like terms: 18.001736027 + -0.25 = 17.751736027 x = 17.751736027 Simplifying x = 17.751736027Subproblem 2
x + 0.25 = -18.001736027 Simplifying x + 0.25 = -18.001736027 Reorder the terms: 0.25 + x = -18.001736027 Solving 0.25 + x = -18.001736027 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 = -18.001736027 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -18.001736027 + -0.25 x = -18.001736027 + -0.25 Combine like terms: -18.001736027 + -0.25 = -18.251736027 x = -18.251736027 Simplifying x = -18.251736027Solution
The solution to the problem is based on the solutions from the subproblems. x = {17.751736027, -18.251736027}
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