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Simplifying x2 + 8x + 22 = 198 + -4x + -0.25x2 Reorder the terms: 22 + 8x + x2 = 198 + -4x + -0.25x2 Solving 22 + 8x + x2 = 198 + -4x + -0.25x2 Solving for variable 'x'. Reorder the terms: 22 + -198 + 8x + 4x + x2 + 0.25x2 = 198 + -4x + -0.25x2 + -198 + 4x + 0.25x2 Combine like terms: 22 + -198 = -176 -176 + 8x + 4x + x2 + 0.25x2 = 198 + -4x + -0.25x2 + -198 + 4x + 0.25x2 Combine like terms: 8x + 4x = 12x -176 + 12x + x2 + 0.25x2 = 198 + -4x + -0.25x2 + -198 + 4x + 0.25x2 Combine like terms: x2 + 0.25x2 = 1.25x2 -176 + 12x + 1.25x2 = 198 + -4x + -0.25x2 + -198 + 4x + 0.25x2 Reorder the terms: -176 + 12x + 1.25x2 = 198 + -198 + -4x + 4x + -0.25x2 + 0.25x2 Combine like terms: 198 + -198 = 0 -176 + 12x + 1.25x2 = 0 + -4x + 4x + -0.25x2 + 0.25x2 -176 + 12x + 1.25x2 = -4x + 4x + -0.25x2 + 0.25x2 Combine like terms: -4x + 4x = 0 -176 + 12x + 1.25x2 = 0 + -0.25x2 + 0.25x2 -176 + 12x + 1.25x2 = -0.25x2 + 0.25x2 Combine like terms: -0.25x2 + 0.25x2 = 0.00 -176 + 12x + 1.25x2 = 0.00 Begin completing the square. Divide all terms by 1.25 the coefficient of the squared term: Divide each side by '1.25'. -140.8 + 9.6x + x2 = 0 Move the constant term to the right: Add '140.8' to each side of the equation. -140.8 + 9.6x + 140.8 + x2 = 0 + 140.8 Reorder the terms: -140.8 + 140.8 + 9.6x + x2 = 0 + 140.8 Combine like terms: -140.8 + 140.8 = 0.0 0.0 + 9.6x + x2 = 0 + 140.8 9.6x + x2 = 0 + 140.8 Combine like terms: 0 + 140.8 = 140.8 9.6x + x2 = 140.8 The x term is 9.6x. Take half its coefficient (4.8). Square it (23.04) and add it to both sides. Add '23.04' to each side of the equation. 9.6x + 23.04 + x2 = 140.8 + 23.04 Reorder the terms: 23.04 + 9.6x + x2 = 140.8 + 23.04 Combine like terms: 140.8 + 23.04 = 163.84 23.04 + 9.6x + x2 = 163.84 Factor a perfect square on the left side: (x + 4.8)(x + 4.8) = 163.84 Calculate the square root of the right side: 12.8 Break this problem into two subproblems by setting (x + 4.8) equal to 12.8 and -12.8.Subproblem 1
x + 4.8 = 12.8 Simplifying x + 4.8 = 12.8 Reorder the terms: 4.8 + x = 12.8 Solving 4.8 + x = 12.8 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.8' to each side of the equation. 4.8 + -4.8 + x = 12.8 + -4.8 Combine like terms: 4.8 + -4.8 = 0.0 0.0 + x = 12.8 + -4.8 x = 12.8 + -4.8 Combine like terms: 12.8 + -4.8 = 8 x = 8 Simplifying x = 8Subproblem 2
x + 4.8 = -12.8 Simplifying x + 4.8 = -12.8 Reorder the terms: 4.8 + x = -12.8 Solving 4.8 + x = -12.8 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.8' to each side of the equation. 4.8 + -4.8 + x = -12.8 + -4.8 Combine like terms: 4.8 + -4.8 = 0.0 0.0 + x = -12.8 + -4.8 x = -12.8 + -4.8 Combine like terms: -12.8 + -4.8 = -17.6 x = -17.6 Simplifying x = -17.6Solution
The solution to the problem is based on the solutions from the subproblems. x = {8, -17.6}
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