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Simplifying 119 = 0.5(X + 3)(X + 6) Reorder the terms: 119 = 0.5(3 + X)(X + 6) Reorder the terms: 119 = 0.5(3 + X)(6 + X) Multiply (3 + X) * (6 + X) 119 = 0.5(3(6 + X) + X(6 + X)) 119 = 0.5((6 * 3 + X * 3) + X(6 + X)) 119 = 0.5((18 + 3X) + X(6 + X)) 119 = 0.5(18 + 3X + (6 * X + X * X)) 119 = 0.5(18 + 3X + (6X + X2)) Combine like terms: 3X + 6X = 9X 119 = 0.5(18 + 9X + X2) 119 = (18 * 0.5 + 9X * 0.5 + X2 * 0.5) 119 = (9 + 4.5X + 0.5X2) Solving 119 = 9 + 4.5X + 0.5X2 Solving for variable 'X'. Combine like terms: 119 + -9 = 110 110 + -4.5X + -0.5X2 = 9 + 4.5X + 0.5X2 + -9 + -4.5X + -0.5X2 Reorder the terms: 110 + -4.5X + -0.5X2 = 9 + -9 + 4.5X + -4.5X + 0.5X2 + -0.5X2 Combine like terms: 9 + -9 = 0 110 + -4.5X + -0.5X2 = 0 + 4.5X + -4.5X + 0.5X2 + -0.5X2 110 + -4.5X + -0.5X2 = 4.5X + -4.5X + 0.5X2 + -0.5X2 Combine like terms: 4.5X + -4.5X = 0.0 110 + -4.5X + -0.5X2 = 0.0 + 0.5X2 + -0.5X2 110 + -4.5X + -0.5X2 = 0.5X2 + -0.5X2 Combine like terms: 0.5X2 + -0.5X2 = 0.0 110 + -4.5X + -0.5X2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -220 + 9X + X2 = 0 Move the constant term to the right: Add '220' to each side of the equation. -220 + 9X + 220 + X2 = 0 + 220 Reorder the terms: -220 + 220 + 9X + X2 = 0 + 220 Combine like terms: -220 + 220 = 0 0 + 9X + X2 = 0 + 220 9X + X2 = 0 + 220 Combine like terms: 0 + 220 = 220 9X + X2 = 220 The X term is 9X. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9X + 20.25 + X2 = 220 + 20.25 Reorder the terms: 20.25 + 9X + X2 = 220 + 20.25 Combine like terms: 220 + 20.25 = 240.25 20.25 + 9X + X2 = 240.25 Factor a perfect square on the left side: (X + 4.5)(X + 4.5) = 240.25 Calculate the square root of the right side: 15.5 Break this problem into two subproblems by setting (X + 4.5) equal to 15.5 and -15.5.Subproblem 1
X + 4.5 = 15.5 Simplifying X + 4.5 = 15.5 Reorder the terms: 4.5 + X = 15.5 Solving 4.5 + X = 15.5 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + X = 15.5 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + X = 15.5 + -4.5 X = 15.5 + -4.5 Combine like terms: 15.5 + -4.5 = 11 X = 11 Simplifying X = 11Subproblem 2
X + 4.5 = -15.5 Simplifying X + 4.5 = -15.5 Reorder the terms: 4.5 + X = -15.5 Solving 4.5 + X = -15.5 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + X = -15.5 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + X = -15.5 + -4.5 X = -15.5 + -4.5 Combine like terms: -15.5 + -4.5 = -20 X = -20 Simplifying X = -20Solution
The solution to the problem is based on the solutions from the subproblems. X = {11, -20}
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