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Simplifying 5135.5 = -3.5x2 + 364x + -965 Reorder the terms: 5135.5 = -965 + 364x + -3.5x2 Solving 5135.5 = -965 + 364x + -3.5x2 Solving for variable 'x'. Combine like terms: 5135.5 + 965 = 6100.5 6100.5 + -364x + 3.5x2 = -965 + 364x + -3.5x2 + 965 + -364x + 3.5x2 Reorder the terms: 6100.5 + -364x + 3.5x2 = -965 + 965 + 364x + -364x + -3.5x2 + 3.5x2 Combine like terms: -965 + 965 = 0 6100.5 + -364x + 3.5x2 = 0 + 364x + -364x + -3.5x2 + 3.5x2 6100.5 + -364x + 3.5x2 = 364x + -364x + -3.5x2 + 3.5x2 Combine like terms: 364x + -364x = 0 6100.5 + -364x + 3.5x2 = 0 + -3.5x2 + 3.5x2 6100.5 + -364x + 3.5x2 = -3.5x2 + 3.5x2 Combine like terms: -3.5x2 + 3.5x2 = 0.0 6100.5 + -364x + 3.5x2 = 0.0 Begin completing the square. Divide all terms by 3.5 the coefficient of the squared term: Divide each side by '3.5'. 1743 + -104x + x2 = 0 Move the constant term to the right: Add '-1743' to each side of the equation. 1743 + -104x + -1743 + x2 = 0 + -1743 Reorder the terms: 1743 + -1743 + -104x + x2 = 0 + -1743 Combine like terms: 1743 + -1743 = 0 0 + -104x + x2 = 0 + -1743 -104x + x2 = 0 + -1743 Combine like terms: 0 + -1743 = -1743 -104x + x2 = -1743 The x term is -104x. Take half its coefficient (-52). Square it (2704) and add it to both sides. Add '2704' to each side of the equation. -104x + 2704 + x2 = -1743 + 2704 Reorder the terms: 2704 + -104x + x2 = -1743 + 2704 Combine like terms: -1743 + 2704 = 961 2704 + -104x + x2 = 961 Factor a perfect square on the left side: (x + -52)(x + -52) = 961 Calculate the square root of the right side: 31 Break this problem into two subproblems by setting (x + -52) equal to 31 and -31.Subproblem 1
x + -52 = 31 Simplifying x + -52 = 31 Reorder the terms: -52 + x = 31 Solving -52 + x = 31 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '52' to each side of the equation. -52 + 52 + x = 31 + 52 Combine like terms: -52 + 52 = 0 0 + x = 31 + 52 x = 31 + 52 Combine like terms: 31 + 52 = 83 x = 83 Simplifying x = 83Subproblem 2
x + -52 = -31 Simplifying x + -52 = -31 Reorder the terms: -52 + x = -31 Solving -52 + x = -31 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '52' to each side of the equation. -52 + 52 + x = -31 + 52 Combine like terms: -52 + 52 = 0 0 + x = -31 + 52 x = -31 + 52 Combine like terms: -31 + 52 = 21 x = 21 Simplifying x = 21Solution
The solution to the problem is based on the solutions from the subproblems. x = {83, 21}
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