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Simplifying 0 = -16x2 + 0.5x + 3 Reorder the terms: 0 = 3 + 0.5x + -16x2 Solving 0 = 3 + 0.5x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -3 = -3 -3 + -0.5x + 16x2 = 3 + 0.5x + -16x2 + -3 + -0.5x + 16x2 Reorder the terms: -3 + -0.5x + 16x2 = 3 + -3 + 0.5x + -0.5x + -16x2 + 16x2 Combine like terms: 3 + -3 = 0 -3 + -0.5x + 16x2 = 0 + 0.5x + -0.5x + -16x2 + 16x2 -3 + -0.5x + 16x2 = 0.5x + -0.5x + -16x2 + 16x2 Combine like terms: 0.5x + -0.5x = 0.0 -3 + -0.5x + 16x2 = 0.0 + -16x2 + 16x2 -3 + -0.5x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -3 + -0.5x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -0.03125x + x2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -0.03125x + 0.1875 + x2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -0.03125x + x2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -0.03125x + x2 = 0 + 0.1875 -0.03125x + x2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -0.03125x + x2 = 0.1875 The x term is -0.03125x. Take half its coefficient (-0.015625). Square it (0.000244140625) and add it to both sides. Add '0.000244140625' to each side of the equation. -0.03125x + 0.000244140625 + x2 = 0.1875 + 0.000244140625 Reorder the terms: 0.000244140625 + -0.03125x + x2 = 0.1875 + 0.000244140625 Combine like terms: 0.1875 + 0.000244140625 = 0.187744140625 0.000244140625 + -0.03125x + x2 = 0.187744140625 Factor a perfect square on the left side: (x + -0.015625)(x + -0.015625) = 0.187744140625 Calculate the square root of the right side: 0.433294519 Break this problem into two subproblems by setting (x + -0.015625) equal to 0.433294519 and -0.433294519.Subproblem 1
x + -0.015625 = 0.433294519 Simplifying x + -0.015625 = 0.433294519 Reorder the terms: -0.015625 + x = 0.433294519 Solving -0.015625 + x = 0.433294519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.015625' to each side of the equation. -0.015625 + 0.015625 + x = 0.433294519 + 0.015625 Combine like terms: -0.015625 + 0.015625 = 0.000000 0.000000 + x = 0.433294519 + 0.015625 x = 0.433294519 + 0.015625 Combine like terms: 0.433294519 + 0.015625 = 0.448919519 x = 0.448919519 Simplifying x = 0.448919519Subproblem 2
x + -0.015625 = -0.433294519 Simplifying x + -0.015625 = -0.433294519 Reorder the terms: -0.015625 + x = -0.433294519 Solving -0.015625 + x = -0.433294519 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.015625' to each side of the equation. -0.015625 + 0.015625 + x = -0.433294519 + 0.015625 Combine like terms: -0.015625 + 0.015625 = 0.000000 0.000000 + x = -0.433294519 + 0.015625 x = -0.433294519 + 0.015625 Combine like terms: -0.433294519 + 0.015625 = -0.417669519 x = -0.417669519 Simplifying x = -0.417669519Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.448919519, -0.417669519}
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