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Simplifying 16x(2 + -1x) + (4x + -5) * 2 = 1 (2 * 16x + -1x * 16x) + (4x + -5) * 2 = 1 (32x + -16x2) + (4x + -5) * 2 = 1 Reorder the terms: 32x + -16x2 + (-5 + 4x) * 2 = 1 Reorder the terms for easier multiplication: 32x + -16x2 + 2(-5 + 4x) = 1 32x + -16x2 + (-5 * 2 + 4x * 2) = 1 32x + -16x2 + (-10 + 8x) = 1 Reorder the terms: -10 + 32x + 8x + -16x2 = 1 Combine like terms: 32x + 8x = 40x -10 + 40x + -16x2 = 1 Solving -10 + 40x + -16x2 = 1 Solving for variable 'x'. Reorder the terms: -10 + -1 + 40x + -16x2 = 1 + -1 Combine like terms: -10 + -1 = -11 -11 + 40x + -16x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -11 + 40x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.6875 + -2.5x + x2 = 0 Move the constant term to the right: Add '-0.6875' to each side of the equation. 0.6875 + -2.5x + -0.6875 + x2 = 0 + -0.6875 Reorder the terms: 0.6875 + -0.6875 + -2.5x + x2 = 0 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + -2.5x + x2 = 0 + -0.6875 -2.5x + x2 = 0 + -0.6875 Combine like terms: 0 + -0.6875 = -0.6875 -2.5x + x2 = -0.6875 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 = -0.6875 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = -0.6875 + 1.5625 Combine like terms: -0.6875 + 1.5625 = 0.875 1.5625 + -2.5x + x2 = 0.875 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 0.875 Calculate the square root of the right side: 0.935414347 Break this problem into two subproblems by setting (x + -1.25) equal to 0.935414347 and -0.935414347.Subproblem 1
x + -1.25 = 0.935414347 Simplifying x + -1.25 = 0.935414347 Reorder the terms: -1.25 + x = 0.935414347 Solving -1.25 + x = 0.935414347 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 = 0.935414347 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.935414347 + 1.25 x = 0.935414347 + 1.25 Combine like terms: 0.935414347 + 1.25 = 2.185414347 x = 2.185414347 Simplifying x = 2.185414347Subproblem 2
x + -1.25 = -0.935414347 Simplifying x + -1.25 = -0.935414347 Reorder the terms: -1.25 + x = -0.935414347 Solving -1.25 + x = -0.935414347 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 = -0.935414347 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.935414347 + 1.25 x = -0.935414347 + 1.25 Combine like terms: -0.935414347 + 1.25 = 0.314585653 x = 0.314585653 Simplifying x = 0.314585653Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.185414347, 0.314585653}
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