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Simplifying -9f2 + -8f + 7 = 0 Reorder the terms: 7 + -8f + -9f2 = 0 Solving 7 + -8f + -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'. -0.7777777778 + 0.8888888889f + f2 = 0 Move the constant term to the right: Add '0.7777777778' to each side of the equation. -0.7777777778 + 0.8888888889f + 0.7777777778 + f2 = 0 + 0.7777777778 Reorder the terms: -0.7777777778 + 0.7777777778 + 0.8888888889f + f2 = 0 + 0.7777777778 Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000 0.0000000000 + 0.8888888889f + f2 = 0 + 0.7777777778 0.8888888889f + f2 = 0 + 0.7777777778 Combine like terms: 0 + 0.7777777778 = 0.7777777778 0.8888888889f + f2 = 0.7777777778 The f term is 0.8888888889f. Take half its coefficient (0.4444444445). Square it (0.1975308642) and add it to both sides. Add '0.1975308642' to each side of the equation. 0.8888888889f + 0.1975308642 + f2 = 0.7777777778 + 0.1975308642 Reorder the terms: 0.1975308642 + 0.8888888889f + f2 = 0.7777777778 + 0.1975308642 Combine like terms: 0.7777777778 + 0.1975308642 = 0.975308642 0.1975308642 + 0.8888888889f + f2 = 0.975308642 Factor a perfect square on the left side: (f + 0.4444444445)(f + 0.4444444445) = 0.975308642 Calculate the square root of the right side: 0.987577157 Break this problem into two subproblems by setting (f + 0.4444444445) equal to 0.987577157 and -0.987577157.Subproblem 1
f + 0.4444444445 = 0.987577157 Simplifying f + 0.4444444445 = 0.987577157 Reorder the terms: 0.4444444445 + f = 0.987577157 Solving 0.4444444445 + f = 0.987577157 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + f = 0.987577157 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + f = 0.987577157 + -0.4444444445 f = 0.987577157 + -0.4444444445 Combine like terms: 0.987577157 + -0.4444444445 = 0.5431327125 f = 0.5431327125 Simplifying f = 0.5431327125Subproblem 2
f + 0.4444444445 = -0.987577157 Simplifying f + 0.4444444445 = -0.987577157 Reorder the terms: 0.4444444445 + f = -0.987577157 Solving 0.4444444445 + f = -0.987577157 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.4444444445' to each side of the equation. 0.4444444445 + -0.4444444445 + f = -0.987577157 + -0.4444444445 Combine like terms: 0.4444444445 + -0.4444444445 = 0.0000000000 0.0000000000 + f = -0.987577157 + -0.4444444445 f = -0.987577157 + -0.4444444445 Combine like terms: -0.987577157 + -0.4444444445 = -1.4320216015 f = -1.4320216015 Simplifying f = -1.4320216015Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.5431327125, -1.4320216015}
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