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Simplifying 8x(4x + 7) = 1 Reorder the terms: 8x(7 + 4x) = 1 (7 * 8x + 4x * 8x) = 1 (56x + 32x2) = 1 Solving 56x + 32x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 56x + 32x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 56x + 32x2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.03125 + 1.75x + x2 = 0 Move the constant term to the right: Add '0.03125' to each side of the equation. -0.03125 + 1.75x + 0.03125 + x2 = 0 + 0.03125 Reorder the terms: -0.03125 + 0.03125 + 1.75x + x2 = 0 + 0.03125 Combine like terms: -0.03125 + 0.03125 = 0.00000 0.00000 + 1.75x + x2 = 0 + 0.03125 1.75x + x2 = 0 + 0.03125 Combine like terms: 0 + 0.03125 = 0.03125 1.75x + x2 = 0.03125 The x term is 1.75x. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75x + 0.765625 + x2 = 0.03125 + 0.765625 Reorder the terms: 0.765625 + 1.75x + x2 = 0.03125 + 0.765625 Combine like terms: 0.03125 + 0.765625 = 0.796875 0.765625 + 1.75x + x2 = 0.796875 Factor a perfect square on the left side: (x + 0.875)(x + 0.875) = 0.796875 Calculate the square root of the right side: 0.892678554 Break this problem into two subproblems by setting (x + 0.875) equal to 0.892678554 and -0.892678554.Subproblem 1
x + 0.875 = 0.892678554 Simplifying x + 0.875 = 0.892678554 Reorder the terms: 0.875 + x = 0.892678554 Solving 0.875 + x = 0.892678554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = 0.892678554 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = 0.892678554 + -0.875 x = 0.892678554 + -0.875 Combine like terms: 0.892678554 + -0.875 = 0.017678554 x = 0.017678554 Simplifying x = 0.017678554Subproblem 2
x + 0.875 = -0.892678554 Simplifying x + 0.875 = -0.892678554 Reorder the terms: 0.875 + x = -0.892678554 Solving 0.875 + x = -0.892678554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + x = -0.892678554 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + x = -0.892678554 + -0.875 x = -0.892678554 + -0.875 Combine like terms: -0.892678554 + -0.875 = -1.767678554 x = -1.767678554 Simplifying x = -1.767678554Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.017678554, -1.767678554}
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