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Simplifying 0.25x + 0.50x(67 + -1x) = 22.25 0.25x + (67 * 0.50x + -1x * 0.50x) = 22.25 0.25x + (33.5x + -0.5x2) = 22.25 Combine like terms: 0.25x + 33.5x = 33.75x 33.75x + -0.5x2 = 22.25 Solving 33.75x + -0.5x2 = 22.25 Solving for variable 'x'. Reorder the terms: -22.25 + 33.75x + -0.5x2 = 22.25 + -22.25 Combine like terms: 22.25 + -22.25 = 0.00 -22.25 + 33.75x + -0.5x2 = 0.00 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 44.5 + -67.5x + x2 = 0.0 Move the constant term to the right: Add '-44.5' to each side of the equation. 44.5 + -67.5x + -44.5 + x2 = 0.0 + -44.5 Reorder the terms: 44.5 + -44.5 + -67.5x + x2 = 0.0 + -44.5 Combine like terms: 44.5 + -44.5 = 0.0 0.0 + -67.5x + x2 = 0.0 + -44.5 -67.5x + x2 = 0.0 + -44.5 Combine like terms: 0.0 + -44.5 = -44.5 -67.5x + x2 = -44.5 The x term is -67.5x. Take half its coefficient (-33.75). Square it (1139.0625) and add it to both sides. Add '1139.0625' to each side of the equation. -67.5x + 1139.0625 + x2 = -44.5 + 1139.0625 Reorder the terms: 1139.0625 + -67.5x + x2 = -44.5 + 1139.0625 Combine like terms: -44.5 + 1139.0625 = 1094.5625 1139.0625 + -67.5x + x2 = 1094.5625 Factor a perfect square on the left side: (x + -33.75)(x + -33.75) = 1094.5625 Calculate the square root of the right side: 33.084172953 Break this problem into two subproblems by setting (x + -33.75) equal to 33.084172953 and -33.084172953.Subproblem 1
x + -33.75 = 33.084172953 Simplifying x + -33.75 = 33.084172953 Reorder the terms: -33.75 + x = 33.084172953 Solving -33.75 + x = 33.084172953 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '33.75' to each side of the equation. -33.75 + 33.75 + x = 33.084172953 + 33.75 Combine like terms: -33.75 + 33.75 = 0.00 0.00 + x = 33.084172953 + 33.75 x = 33.084172953 + 33.75 Combine like terms: 33.084172953 + 33.75 = 66.834172953 x = 66.834172953 Simplifying x = 66.834172953Subproblem 2
x + -33.75 = -33.084172953 Simplifying x + -33.75 = -33.084172953 Reorder the terms: -33.75 + x = -33.084172953 Solving -33.75 + x = -33.084172953 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '33.75' to each side of the equation. -33.75 + 33.75 + x = -33.084172953 + 33.75 Combine like terms: -33.75 + 33.75 = 0.00 0.00 + x = -33.084172953 + 33.75 x = -33.084172953 + 33.75 Combine like terms: -33.084172953 + 33.75 = 0.665827047 x = 0.665827047 Simplifying x = 0.665827047Solution
The solution to the problem is based on the solutions from the subproblems. x = {66.834172953, 0.665827047}
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