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Simplifying 16x2 + 8x + 1 = 24 Reorder the terms: 1 + 8x + 16x2 = 24 Solving 1 + 8x + 16x2 = 24 Solving for variable 'x'. Reorder the terms: 1 + -24 + 8x + 16x2 = 24 + -24 Combine like terms: 1 + -24 = -23 -23 + 8x + 16x2 = 24 + -24 Combine like terms: 24 + -24 = 0 -23 + 8x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.4375 + 0.5x + x2 = 0 Move the constant term to the right: Add '1.4375' to each side of the equation. -1.4375 + 0.5x + 1.4375 + x2 = 0 + 1.4375 Reorder the terms: -1.4375 + 1.4375 + 0.5x + x2 = 0 + 1.4375 Combine like terms: -1.4375 + 1.4375 = 0.0000 0.0000 + 0.5x + x2 = 0 + 1.4375 0.5x + x2 = 0 + 1.4375 Combine like terms: 0 + 1.4375 = 1.4375 0.5x + x2 = 1.4375 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 = 1.4375 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 1.4375 + 0.0625 Combine like terms: 1.4375 + 0.0625 = 1.5 0.0625 + 0.5x + x2 = 1.5 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (x + 0.25) equal to 1.224744871 and -1.224744871.Subproblem 1
x + 0.25 = 1.224744871 Simplifying x + 0.25 = 1.224744871 Reorder the terms: 0.25 + x = 1.224744871 Solving 0.25 + x = 1.224744871 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 = 1.224744871 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.224744871 + -0.25 x = 1.224744871 + -0.25 Combine like terms: 1.224744871 + -0.25 = 0.974744871 x = 0.974744871 Simplifying x = 0.974744871Subproblem 2
x + 0.25 = -1.224744871 Simplifying x + 0.25 = -1.224744871 Reorder the terms: 0.25 + x = -1.224744871 Solving 0.25 + x = -1.224744871 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 = -1.224744871 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.224744871 + -0.25 x = -1.224744871 + -0.25 Combine like terms: -1.224744871 + -0.25 = -1.474744871 x = -1.474744871 Simplifying x = -1.474744871Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.974744871, -1.474744871}
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