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Simplifying -0.2x2 + 178x = 78x + 12320 Reorder the terms: 178x + -0.2x2 = 78x + 12320 Reorder the terms: 178x + -0.2x2 = 12320 + 78x Solving 178x + -0.2x2 = 12320 + 78x Solving for variable 'x'. Reorder the terms: -12320 + 178x + -78x + -0.2x2 = 12320 + 78x + -12320 + -78x Combine like terms: 178x + -78x = 100x -12320 + 100x + -0.2x2 = 12320 + 78x + -12320 + -78x Reorder the terms: -12320 + 100x + -0.2x2 = 12320 + -12320 + 78x + -78x Combine like terms: 12320 + -12320 = 0 -12320 + 100x + -0.2x2 = 0 + 78x + -78x -12320 + 100x + -0.2x2 = 78x + -78x Combine like terms: 78x + -78x = 0 -12320 + 100x + -0.2x2 = 0 Begin completing the square. Divide all terms by -0.2 the coefficient of the squared term: Divide each side by '-0.2'. 61600 + -500x + x2 = 0 Move the constant term to the right: Add '-61600' to each side of the equation. 61600 + -500x + -61600 + x2 = 0 + -61600 Reorder the terms: 61600 + -61600 + -500x + x2 = 0 + -61600 Combine like terms: 61600 + -61600 = 0 0 + -500x + x2 = 0 + -61600 -500x + x2 = 0 + -61600 Combine like terms: 0 + -61600 = -61600 -500x + x2 = -61600 The x term is -500x. Take half its coefficient (-250). Square it (62500) and add it to both sides. Add '62500' to each side of the equation. -500x + 62500 + x2 = -61600 + 62500 Reorder the terms: 62500 + -500x + x2 = -61600 + 62500 Combine like terms: -61600 + 62500 = 900 62500 + -500x + x2 = 900 Factor a perfect square on the left side: (x + -250)(x + -250) = 900 Calculate the square root of the right side: 30 Break this problem into two subproblems by setting (x + -250) equal to 30 and -30.Subproblem 1
x + -250 = 30 Simplifying x + -250 = 30 Reorder the terms: -250 + x = 30 Solving -250 + x = 30 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '250' to each side of the equation. -250 + 250 + x = 30 + 250 Combine like terms: -250 + 250 = 0 0 + x = 30 + 250 x = 30 + 250 Combine like terms: 30 + 250 = 280 x = 280 Simplifying x = 280Subproblem 2
x + -250 = -30 Simplifying x + -250 = -30 Reorder the terms: -250 + x = -30 Solving -250 + x = -30 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '250' to each side of the equation. -250 + 250 + x = -30 + 250 Combine like terms: -250 + 250 = 0 0 + x = -30 + 250 x = -30 + 250 Combine like terms: -30 + 250 = 220 x = 220 Simplifying x = 220Solution
The solution to the problem is based on the solutions from the subproblems. x = {280, 220}
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