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Simplifying 8f2 + 8f + -1 = 0 Reorder the terms: -1 + 8f + 8f2 = 0 Solving -1 + 8f + 8f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + f + f2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + f + 0.125 + f2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + f + f2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + f + f2 = 0 + 0.125 f + f2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 f + f2 = 0.125 The f term is f. 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 + f2 = 0.125 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + f2 = 0.125 + 0.25 Combine like terms: 0.125 + 0.25 = 0.375 1.25 + f2 = 0.375 Factor a perfect square on the left side: (f + 0.5)(f + 0.5) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (f + 0.5) equal to 0.612372436 and -0.612372436.Subproblem 1
f + 0.5 = 0.612372436 Simplifying f + 0.5 = 0.612372436 Reorder the terms: 0.5 + f = 0.612372436 Solving 0.5 + f = 0.612372436 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + f = 0.612372436 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + f = 0.612372436 + -0.5 f = 0.612372436 + -0.5 Combine like terms: 0.612372436 + -0.5 = 0.112372436 f = 0.112372436 Simplifying f = 0.112372436Subproblem 2
f + 0.5 = -0.612372436 Simplifying f + 0.5 = -0.612372436 Reorder the terms: 0.5 + f = -0.612372436 Solving 0.5 + f = -0.612372436 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + f = -0.612372436 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + f = -0.612372436 + -0.5 f = -0.612372436 + -0.5 Combine like terms: -0.612372436 + -0.5 = -1.112372436 f = -1.112372436 Simplifying f = -1.112372436Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.112372436, -1.112372436}
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