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Simplifying 8(1 + 5x) * x + 5 = 13 + 5x Reorder the terms for easier multiplication: 8x(1 + 5x) + 5 = 13 + 5x (1 * 8x + 5x * 8x) + 5 = 13 + 5x (8x + 40x2) + 5 = 13 + 5x Reorder the terms: 5 + 8x + 40x2 = 13 + 5x Solving 5 + 8x + 40x2 = 13 + 5x Solving for variable 'x'. Reorder the terms: 5 + -13 + 8x + -5x + 40x2 = 13 + 5x + -13 + -5x Combine like terms: 5 + -13 = -8 -8 + 8x + -5x + 40x2 = 13 + 5x + -13 + -5x Combine like terms: 8x + -5x = 3x -8 + 3x + 40x2 = 13 + 5x + -13 + -5x Reorder the terms: -8 + 3x + 40x2 = 13 + -13 + 5x + -5x Combine like terms: 13 + -13 = 0 -8 + 3x + 40x2 = 0 + 5x + -5x -8 + 3x + 40x2 = 5x + -5x Combine like terms: 5x + -5x = 0 -8 + 3x + 40x2 = 0 Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -0.2 + 0.075x + x2 = 0.0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 0.075x + 0.2 + x2 = 0.0 + 0.2 Reorder the terms: -0.2 + 0.2 + 0.075x + x2 = 0.0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 0.075x + x2 = 0.0 + 0.2 0.075x + x2 = 0.0 + 0.2 Combine like terms: 0.0 + 0.2 = 0.2 0.075x + x2 = 0.2 The x term is 0.075x. Take half its coefficient (0.0375). Square it (0.00140625) and add it to both sides. Add '0.00140625' to each side of the equation. 0.075x + 0.00140625 + x2 = 0.2 + 0.00140625 Reorder the terms: 0.00140625 + 0.075x + x2 = 0.2 + 0.00140625 Combine like terms: 0.2 + 0.00140625 = 0.20140625 0.00140625 + 0.075x + x2 = 0.20140625 Factor a perfect square on the left side: (x + 0.0375)(x + 0.0375) = 0.20140625 Calculate the square root of the right side: 0.448783077 Break this problem into two subproblems by setting (x + 0.0375) equal to 0.448783077 and -0.448783077.Subproblem 1
x + 0.0375 = 0.448783077 Simplifying x + 0.0375 = 0.448783077 Reorder the terms: 0.0375 + x = 0.448783077 Solving 0.0375 + x = 0.448783077 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0375' to each side of the equation. 0.0375 + -0.0375 + x = 0.448783077 + -0.0375 Combine like terms: 0.0375 + -0.0375 = 0.0000 0.0000 + x = 0.448783077 + -0.0375 x = 0.448783077 + -0.0375 Combine like terms: 0.448783077 + -0.0375 = 0.411283077 x = 0.411283077 Simplifying x = 0.411283077Subproblem 2
x + 0.0375 = -0.448783077 Simplifying x + 0.0375 = -0.448783077 Reorder the terms: 0.0375 + x = -0.448783077 Solving 0.0375 + x = -0.448783077 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0375' to each side of the equation. 0.0375 + -0.0375 + x = -0.448783077 + -0.0375 Combine like terms: 0.0375 + -0.0375 = 0.0000 0.0000 + x = -0.448783077 + -0.0375 x = -0.448783077 + -0.0375 Combine like terms: -0.448783077 + -0.0375 = -0.486283077 x = -0.486283077 Simplifying x = -0.486283077Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.411283077, -0.486283077}
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