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Simplifying x2 + 4x2 + 1x + 1 = 461 Reorder the terms: 1 + 1x + x2 + 4x2 = 461 Combine like terms: x2 + 4x2 = 5x2 1 + 1x + 5x2 = 461 Solving 1 + 1x + 5x2 = 461 Solving for variable 'x'. Reorder the terms: 1 + -461 + 1x + 5x2 = 461 + -461 Combine like terms: 1 + -461 = -460 -460 + 1x + 5x2 = 461 + -461 Combine like terms: 461 + -461 = 0 -460 + 1x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -92 + 0.2x + x2 = 0 Move the constant term to the right: Add '92' to each side of the equation. -92 + 0.2x + 92 + x2 = 0 + 92 Reorder the terms: -92 + 92 + 0.2x + x2 = 0 + 92 Combine like terms: -92 + 92 = 0 0 + 0.2x + x2 = 0 + 92 0.2x + x2 = 0 + 92 Combine like terms: 0 + 92 = 92 0.2x + x2 = 92 The x term is 0.2x. Take half its coefficient (0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. 0.2x + 0.01 + x2 = 92 + 0.01 Reorder the terms: 0.01 + 0.2x + x2 = 92 + 0.01 Combine like terms: 92 + 0.01 = 92.01 0.01 + 0.2x + x2 = 92.01 Factor a perfect square on the left side: (x + 0.1)(x + 0.1) = 92.01 Calculate the square root of the right side: 9.592184318 Break this problem into two subproblems by setting (x + 0.1) equal to 9.592184318 and -9.592184318.Subproblem 1
x + 0.1 = 9.592184318 Simplifying x + 0.1 = 9.592184318 Reorder the terms: 0.1 + x = 9.592184318 Solving 0.1 + x = 9.592184318 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = 9.592184318 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = 9.592184318 + -0.1 x = 9.592184318 + -0.1 Combine like terms: 9.592184318 + -0.1 = 9.492184318 x = 9.492184318 Simplifying x = 9.492184318Subproblem 2
x + 0.1 = -9.592184318 Simplifying x + 0.1 = -9.592184318 Reorder the terms: 0.1 + x = -9.592184318 Solving 0.1 + x = -9.592184318 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = -9.592184318 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = -9.592184318 + -0.1 x = -9.592184318 + -0.1 Combine like terms: -9.592184318 + -0.1 = -9.692184318 x = -9.692184318 Simplifying x = -9.692184318Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.492184318, -9.692184318}
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