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Simplifying (2f + -3) * f = 18 Reorder the terms: (-3 + 2f) * f = 18 Reorder the terms for easier multiplication: f(-3 + 2f) = 18 (-3 * f + 2f * f) = 18 (-3f + 2f2) = 18 Solving -3f + 2f2 = 18 Solving for variable 'f'. Reorder the terms: -18 + -3f + 2f2 = 18 + -18 Combine like terms: 18 + -18 = 0 -18 + -3f + 2f2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9 + -1.5f + f2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -1.5f + 9 + f2 = 0 + 9 Reorder the terms: -9 + 9 + -1.5f + f2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -1.5f + f2 = 0 + 9 -1.5f + f2 = 0 + 9 Combine like terms: 0 + 9 = 9 -1.5f + f2 = 9 The f term is -1.5f. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5f + 0.5625 + f2 = 9 + 0.5625 Reorder the terms: 0.5625 + -1.5f + f2 = 9 + 0.5625 Combine like terms: 9 + 0.5625 = 9.5625 0.5625 + -1.5f + f2 = 9.5625 Factor a perfect square on the left side: (f + -0.75)(f + -0.75) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (f + -0.75) equal to 3.092329219 and -3.092329219.Subproblem 1
f + -0.75 = 3.092329219 Simplifying f + -0.75 = 3.092329219 Reorder the terms: -0.75 + f = 3.092329219 Solving -0.75 + f = 3.092329219 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + f = 3.092329219 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + f = 3.092329219 + 0.75 f = 3.092329219 + 0.75 Combine like terms: 3.092329219 + 0.75 = 3.842329219 f = 3.842329219 Simplifying f = 3.842329219Subproblem 2
f + -0.75 = -3.092329219 Simplifying f + -0.75 = -3.092329219 Reorder the terms: -0.75 + f = -3.092329219 Solving -0.75 + f = -3.092329219 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + f = -3.092329219 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + f = -3.092329219 + 0.75 f = -3.092329219 + 0.75 Combine like terms: -3.092329219 + 0.75 = -2.342329219 f = -2.342329219 Simplifying f = -2.342329219Solution
The solution to the problem is based on the solutions from the subproblems. f = {3.842329219, -2.342329219}
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