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Simplifying -8f2 + 9f + 3 = 0 Reorder the terms: 3 + 9f + -8f2 = 0 Solving 3 + 9f + -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'. -0.375 + -1.125f + f2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -1.125f + 0.375 + f2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -1.125f + f2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -1.125f + f2 = 0 + 0.375 -1.125f + f2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -1.125f + f2 = 0.375 The f term is -1.125f. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125f + 0.31640625 + f2 = 0.375 + 0.31640625 Reorder the terms: 0.31640625 + -1.125f + f2 = 0.375 + 0.31640625 Combine like terms: 0.375 + 0.31640625 = 0.69140625 0.31640625 + -1.125f + f2 = 0.69140625 Factor a perfect square on the left side: (f + -0.5625)(f + -0.5625) = 0.69140625 Calculate the square root of the right side: 0.831508418 Break this problem into two subproblems by setting (f + -0.5625) equal to 0.831508418 and -0.831508418.Subproblem 1
f + -0.5625 = 0.831508418 Simplifying f + -0.5625 = 0.831508418 Reorder the terms: -0.5625 + f = 0.831508418 Solving -0.5625 + f = 0.831508418 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + f = 0.831508418 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + f = 0.831508418 + 0.5625 f = 0.831508418 + 0.5625 Combine like terms: 0.831508418 + 0.5625 = 1.394008418 f = 1.394008418 Simplifying f = 1.394008418Subproblem 2
f + -0.5625 = -0.831508418 Simplifying f + -0.5625 = -0.831508418 Reorder the terms: -0.5625 + f = -0.831508418 Solving -0.5625 + f = -0.831508418 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + f = -0.831508418 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + f = -0.831508418 + 0.5625 f = -0.831508418 + 0.5625 Combine like terms: -0.831508418 + 0.5625 = -0.269008418 f = -0.269008418 Simplifying f = -0.269008418Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.394008418, -0.269008418}
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