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Simplifying 64x2 + 64x + 11 = 0 Reorder the terms: 11 + 64x + 64x2 = 0 Solving 11 + 64x + 64x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. 0.171875 + x + x2 = 0 Move the constant term to the right: Add '-0.171875' to each side of the equation. 0.171875 + x + -0.171875 + x2 = 0 + -0.171875 Reorder the terms: 0.171875 + -0.171875 + x + x2 = 0 + -0.171875 Combine like terms: 0.171875 + -0.171875 = 0.000000 0.000000 + x + x2 = 0 + -0.171875 x + x2 = 0 + -0.171875 Combine like terms: 0 + -0.171875 = -0.171875 x + x2 = -0.171875 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = -0.171875 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = -0.171875 + 0.25 Combine like terms: -0.171875 + 0.25 = 0.078125 1.25 + x2 = 0.078125 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.078125 Calculate the square root of the right side: 0.279508497 Break this problem into two subproblems by setting (x + 0.5) equal to 0.279508497 and -0.279508497.Subproblem 1
x + 0.5 = 0.279508497 Simplifying x + 0.5 = 0.279508497 Reorder the terms: 0.5 + x = 0.279508497 Solving 0.5 + x = 0.279508497 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.279508497 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.279508497 + -0.5 x = 0.279508497 + -0.5 Combine like terms: 0.279508497 + -0.5 = -0.220491503 x = -0.220491503 Simplifying x = -0.220491503Subproblem 2
x + 0.5 = -0.279508497 Simplifying x + 0.5 = -0.279508497 Reorder the terms: 0.5 + x = -0.279508497 Solving 0.5 + x = -0.279508497 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.279508497 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.279508497 + -0.5 x = -0.279508497 + -0.5 Combine like terms: -0.279508497 + -0.5 = -0.779508497 x = -0.779508497 Simplifying x = -0.779508497Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.220491503, -0.779508497}
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