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Simplifying -9f + 15 = -22 + -4f(f + 12) Reorder the terms: 15 + -9f = -22 + -4f(f + 12) Reorder the terms: 15 + -9f = -22 + -4f(12 + f) 15 + -9f = -22 + (12 * -4f + f * -4f) 15 + -9f = -22 + (-48f + -4f2) Solving 15 + -9f = -22 + -48f + -4f2 Solving for variable 'f'. Reorder the terms: 15 + 22 + -9f + 48f + 4f2 = -22 + -48f + -4f2 + 22 + 48f + 4f2 Combine like terms: 15 + 22 = 37 37 + -9f + 48f + 4f2 = -22 + -48f + -4f2 + 22 + 48f + 4f2 Combine like terms: -9f + 48f = 39f 37 + 39f + 4f2 = -22 + -48f + -4f2 + 22 + 48f + 4f2 Reorder the terms: 37 + 39f + 4f2 = -22 + 22 + -48f + 48f + -4f2 + 4f2 Combine like terms: -22 + 22 = 0 37 + 39f + 4f2 = 0 + -48f + 48f + -4f2 + 4f2 37 + 39f + 4f2 = -48f + 48f + -4f2 + 4f2 Combine like terms: -48f + 48f = 0 37 + 39f + 4f2 = 0 + -4f2 + 4f2 37 + 39f + 4f2 = -4f2 + 4f2 Combine like terms: -4f2 + 4f2 = 0 37 + 39f + 4f2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 9.25 + 9.75f + f2 = 0 Move the constant term to the right: Add '-9.25' to each side of the equation. 9.25 + 9.75f + -9.25 + f2 = 0 + -9.25 Reorder the terms: 9.25 + -9.25 + 9.75f + f2 = 0 + -9.25 Combine like terms: 9.25 + -9.25 = 0.00 0.00 + 9.75f + f2 = 0 + -9.25 9.75f + f2 = 0 + -9.25 Combine like terms: 0 + -9.25 = -9.25 9.75f + f2 = -9.25 The f term is 9.75f. Take half its coefficient (4.875). Square it (23.765625) and add it to both sides. Add '23.765625' to each side of the equation. 9.75f + 23.765625 + f2 = -9.25 + 23.765625 Reorder the terms: 23.765625 + 9.75f + f2 = -9.25 + 23.765625 Combine like terms: -9.25 + 23.765625 = 14.515625 23.765625 + 9.75f + f2 = 14.515625 Factor a perfect square on the left side: (f + 4.875)(f + 4.875) = 14.515625 Calculate the square root of the right side: 3.809937664 Break this problem into two subproblems by setting (f + 4.875) equal to 3.809937664 and -3.809937664.Subproblem 1
f + 4.875 = 3.809937664 Simplifying f + 4.875 = 3.809937664 Reorder the terms: 4.875 + f = 3.809937664 Solving 4.875 + f = 3.809937664 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4.875' to each side of the equation. 4.875 + -4.875 + f = 3.809937664 + -4.875 Combine like terms: 4.875 + -4.875 = 0.000 0.000 + f = 3.809937664 + -4.875 f = 3.809937664 + -4.875 Combine like terms: 3.809937664 + -4.875 = -1.065062336 f = -1.065062336 Simplifying f = -1.065062336Subproblem 2
f + 4.875 = -3.809937664 Simplifying f + 4.875 = -3.809937664 Reorder the terms: 4.875 + f = -3.809937664 Solving 4.875 + f = -3.809937664 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4.875' to each side of the equation. 4.875 + -4.875 + f = -3.809937664 + -4.875 Combine like terms: 4.875 + -4.875 = 0.000 0.000 + f = -3.809937664 + -4.875 f = -3.809937664 + -4.875 Combine like terms: -3.809937664 + -4.875 = -8.684937664 f = -8.684937664 Simplifying f = -8.684937664Solution
The solution to the problem is based on the solutions from the subproblems. f = {-1.065062336, -8.684937664}
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