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Simplifying 16x2 + 40x + 23 = 0 Reorder the terms: 23 + 40x + 16x2 = 0 Solving 23 + 40x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 1.4375 + 2.5x + x2 = 0 Move the constant term to the right: Add '-1.4375' to each side of the equation. 1.4375 + 2.5x + -1.4375 + x2 = 0 + -1.4375 Reorder the terms: 1.4375 + -1.4375 + 2.5x + x2 = 0 + -1.4375 Combine like terms: 1.4375 + -1.4375 = 0.0000 0.0000 + 2.5x + x2 = 0 + -1.4375 2.5x + x2 = 0 + -1.4375 Combine like terms: 0 + -1.4375 = -1.4375 2.5x + x2 = -1.4375 The x term is 2.5x. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5x + 1.5625 + x2 = -1.4375 + 1.5625 Reorder the terms: 1.5625 + 2.5x + x2 = -1.4375 + 1.5625 Combine like terms: -1.4375 + 1.5625 = 0.125 1.5625 + 2.5x + x2 = 0.125 Factor a perfect square on the left side: (x + 1.25)(x + 1.25) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (x + 1.25) equal to 0.353553391 and -0.353553391.Subproblem 1
x + 1.25 = 0.353553391 Simplifying x + 1.25 = 0.353553391 Reorder the terms: 1.25 + x = 0.353553391 Solving 1.25 + x = 0.353553391 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 0.353553391 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 0.353553391 + -1.25 x = 0.353553391 + -1.25 Combine like terms: 0.353553391 + -1.25 = -0.896446609 x = -0.896446609 Simplifying x = -0.896446609Subproblem 2
x + 1.25 = -0.353553391 Simplifying x + 1.25 = -0.353553391 Reorder the terms: 1.25 + x = -0.353553391 Solving 1.25 + x = -0.353553391 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -0.353553391 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -0.353553391 + -1.25 x = -0.353553391 + -1.25 Combine like terms: -0.353553391 + -1.25 = -1.603553391 x = -1.603553391 Simplifying x = -1.603553391Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.896446609, -1.603553391}
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