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Simplifying 6f2 + 56f + 30 = 0 Reorder the terms: 30 + 56f + 6f2 = 0 Solving 30 + 56f + 6f2 = 0 Solving for variable 'f'. Factor out the Greatest Common Factor (GCF), '2'. 2(15 + 28f + 3f2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(15 + 28f + 3f2)' equal to zero and attempt to solve: Simplifying 15 + 28f + 3f2 = 0 Solving 15 + 28f + 3f2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 5 + 9.333333333f + f2 = 0 Move the constant term to the right: Add '-5' to each side of the equation. 5 + 9.333333333f + -5 + f2 = 0 + -5 Reorder the terms: 5 + -5 + 9.333333333f + f2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 9.333333333f + f2 = 0 + -5 9.333333333f + f2 = 0 + -5 Combine like terms: 0 + -5 = -5 9.333333333f + f2 = -5 The f term is 9.333333333f. Take half its coefficient (4.666666667). Square it (21.77777778) and add it to both sides. Add '21.77777778' to each side of the equation. 9.333333333f + 21.77777778 + f2 = -5 + 21.77777778 Reorder the terms: 21.77777778 + 9.333333333f + f2 = -5 + 21.77777778 Combine like terms: -5 + 21.77777778 = 16.77777778 21.77777778 + 9.333333333f + f2 = 16.77777778 Factor a perfect square on the left side: (f + 4.666666667)(f + 4.666666667) = 16.77777778 Calculate the square root of the right side: 4.096068576 Break this problem into two subproblems by setting (f + 4.666666667) equal to 4.096068576 and -4.096068576.Subproblem 1
f + 4.666666667 = 4.096068576 Simplifying f + 4.666666667 = 4.096068576 Reorder the terms: 4.666666667 + f = 4.096068576 Solving 4.666666667 + f = 4.096068576 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4.666666667' to each side of the equation. 4.666666667 + -4.666666667 + f = 4.096068576 + -4.666666667 Combine like terms: 4.666666667 + -4.666666667 = 0.000000000 0.000000000 + f = 4.096068576 + -4.666666667 f = 4.096068576 + -4.666666667 Combine like terms: 4.096068576 + -4.666666667 = -0.570598091 f = -0.570598091 Simplifying f = -0.570598091Subproblem 2
f + 4.666666667 = -4.096068576 Simplifying f + 4.666666667 = -4.096068576 Reorder the terms: 4.666666667 + f = -4.096068576 Solving 4.666666667 + f = -4.096068576 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4.666666667' to each side of the equation. 4.666666667 + -4.666666667 + f = -4.096068576 + -4.666666667 Combine like terms: 4.666666667 + -4.666666667 = 0.000000000 0.000000000 + f = -4.096068576 + -4.666666667 f = -4.096068576 + -4.666666667 Combine like terms: -4.096068576 + -4.666666667 = -8.762735243 f = -8.762735243 Simplifying f = -8.762735243Solution
The solution to the problem is based on the solutions from the subproblems. f = {-0.570598091, -8.762735243}Solution
f = {-0.570598091, -8.762735243}
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