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Simplifying 5f + 9f2 + -15 = 0 Reorder the terms: -15 + 5f + 9f2 = 0 Solving -15 + 5f + 9f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.666666667 + 0.5555555556f + f2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 0.5555555556f + 1.666666667 + f2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 0.5555555556f + f2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 0.5555555556f + f2 = 0 + 1.666666667 0.5555555556f + f2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 0.5555555556f + f2 = 1.666666667 The f term is 0.5555555556f. Take half its coefficient (0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. 0.5555555556f + 0.07716049384 + f2 = 1.666666667 + 0.07716049384 Reorder the terms: 0.07716049384 + 0.5555555556f + f2 = 1.666666667 + 0.07716049384 Combine like terms: 1.666666667 + 0.07716049384 = 1.74382716084 0.07716049384 + 0.5555555556f + f2 = 1.74382716084 Factor a perfect square on the left side: (f + 0.2777777778)(f + 0.2777777778) = 1.74382716084 Calculate the square root of the right side: 1.320540481 Break this problem into two subproblems by setting (f + 0.2777777778) equal to 1.320540481 and -1.320540481.Subproblem 1
f + 0.2777777778 = 1.320540481 Simplifying f + 0.2777777778 = 1.320540481 Reorder the terms: 0.2777777778 + f = 1.320540481 Solving 0.2777777778 + f = 1.320540481 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + f = 1.320540481 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + f = 1.320540481 + -0.2777777778 f = 1.320540481 + -0.2777777778 Combine like terms: 1.320540481 + -0.2777777778 = 1.0427627032 f = 1.0427627032 Simplifying f = 1.0427627032Subproblem 2
f + 0.2777777778 = -1.320540481 Simplifying f + 0.2777777778 = -1.320540481 Reorder the terms: 0.2777777778 + f = -1.320540481 Solving 0.2777777778 + f = -1.320540481 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + f = -1.320540481 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + f = -1.320540481 + -0.2777777778 f = -1.320540481 + -0.2777777778 Combine like terms: -1.320540481 + -0.2777777778 = -1.5983182588 f = -1.5983182588 Simplifying f = -1.5983182588Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.0427627032, -1.5983182588}
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