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Simplifying x(2x + 2) = 65 Reorder the terms: x(2 + 2x) = 65 (2 * x + 2x * x) = 65 (2x + 2x2) = 65 Solving 2x + 2x2 = 65 Solving for variable 'x'. Reorder the terms: -65 + 2x + 2x2 = 65 + -65 Combine like terms: 65 + -65 = 0 -65 + 2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -32.5 + x + x2 = 0 Move the constant term to the right: Add '32.5' to each side of the equation. -32.5 + x + 32.5 + x2 = 0 + 32.5 Reorder the terms: -32.5 + 32.5 + x + x2 = 0 + 32.5 Combine like terms: -32.5 + 32.5 = 0.0 0.0 + x + x2 = 0 + 32.5 x + x2 = 0 + 32.5 Combine like terms: 0 + 32.5 = 32.5 x + x2 = 32.5 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 = 32.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 32.5 + 0.25 Combine like terms: 32.5 + 0.25 = 32.75 1.25 + x2 = 32.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 32.75 Calculate the square root of the right side: 5.722761571 Break this problem into two subproblems by setting (x + 0.5) equal to 5.722761571 and -5.722761571.Subproblem 1
x + 0.5 = 5.722761571 Simplifying x + 0.5 = 5.722761571 Reorder the terms: 0.5 + x = 5.722761571 Solving 0.5 + x = 5.722761571 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 = 5.722761571 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 5.722761571 + -0.5 x = 5.722761571 + -0.5 Combine like terms: 5.722761571 + -0.5 = 5.222761571 x = 5.222761571 Simplifying x = 5.222761571Subproblem 2
x + 0.5 = -5.722761571 Simplifying x + 0.5 = -5.722761571 Reorder the terms: 0.5 + x = -5.722761571 Solving 0.5 + x = -5.722761571 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 = -5.722761571 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -5.722761571 + -0.5 x = -5.722761571 + -0.5 Combine like terms: -5.722761571 + -0.5 = -6.222761571 x = -6.222761571 Simplifying x = -6.222761571Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.222761571, -6.222761571}
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