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Simplifying 4f2 + 24f + -76 = 0 Reorder the terms: -76 + 24f + 4f2 = 0 Solving -76 + 24f + 4f2 = 0 Solving for variable 'f'. Factor out the Greatest Common Factor (GCF), '4'. 4(-19 + 6f + f2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-19 + 6f + f2)' equal to zero and attempt to solve: Simplifying -19 + 6f + f2 = 0 Solving -19 + 6f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '19' to each side of the equation. -19 + 6f + 19 + f2 = 0 + 19 Reorder the terms: -19 + 19 + 6f + f2 = 0 + 19 Combine like terms: -19 + 19 = 0 0 + 6f + f2 = 0 + 19 6f + f2 = 0 + 19 Combine like terms: 0 + 19 = 19 6f + f2 = 19 The f term is 6f. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6f + 9 + f2 = 19 + 9 Reorder the terms: 9 + 6f + f2 = 19 + 9 Combine like terms: 19 + 9 = 28 9 + 6f + f2 = 28 Factor a perfect square on the left side: (f + 3)(f + 3) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (f + 3) equal to 5.291502622 and -5.291502622.Subproblem 1
f + 3 = 5.291502622 Simplifying f + 3 = 5.291502622 Reorder the terms: 3 + f = 5.291502622 Solving 3 + f = 5.291502622 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + f = 5.291502622 + -3 Combine like terms: 3 + -3 = 0 0 + f = 5.291502622 + -3 f = 5.291502622 + -3 Combine like terms: 5.291502622 + -3 = 2.291502622 f = 2.291502622 Simplifying f = 2.291502622Subproblem 2
f + 3 = -5.291502622 Simplifying f + 3 = -5.291502622 Reorder the terms: 3 + f = -5.291502622 Solving 3 + f = -5.291502622 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + f = -5.291502622 + -3 Combine like terms: 3 + -3 = 0 0 + f = -5.291502622 + -3 f = -5.291502622 + -3 Combine like terms: -5.291502622 + -3 = -8.291502622 f = -8.291502622 Simplifying f = -8.291502622Solution
The solution to the problem is based on the solutions from the subproblems. f = {2.291502622, -8.291502622}Solution
f = {2.291502622, -8.291502622}
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