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Simplifying 9.7 = -0.01x2 + 0.22x + 8.5 Reorder the terms: 9.7 = 8.5 + 0.22x + -0.01x2 Solving 9.7 = 8.5 + 0.22x + -0.01x2 Solving for variable 'x'. Combine like terms: 9.7 + -8.5 = 1.2 1.2 + -0.22x + 0.01x2 = 8.5 + 0.22x + -0.01x2 + -8.5 + -0.22x + 0.01x2 Reorder the terms: 1.2 + -0.22x + 0.01x2 = 8.5 + -8.5 + 0.22x + -0.22x + -0.01x2 + 0.01x2 Combine like terms: 8.5 + -8.5 = 0.0 1.2 + -0.22x + 0.01x2 = 0.0 + 0.22x + -0.22x + -0.01x2 + 0.01x2 1.2 + -0.22x + 0.01x2 = 0.22x + -0.22x + -0.01x2 + 0.01x2 Combine like terms: 0.22x + -0.22x = 0.00 1.2 + -0.22x + 0.01x2 = 0.00 + -0.01x2 + 0.01x2 1.2 + -0.22x + 0.01x2 = -0.01x2 + 0.01x2 Combine like terms: -0.01x2 + 0.01x2 = 0.00 1.2 + -0.22x + 0.01x2 = 0.00 Begin completing the square. Divide all terms by 0.01 the coefficient of the squared term: Divide each side by '0.01'. 120 + -22x + x2 = 0 Move the constant term to the right: Add '-120' to each side of the equation. 120 + -22x + -120 + x2 = 0 + -120 Reorder the terms: 120 + -120 + -22x + x2 = 0 + -120 Combine like terms: 120 + -120 = 0 0 + -22x + x2 = 0 + -120 -22x + x2 = 0 + -120 Combine like terms: 0 + -120 = -120 -22x + x2 = -120 The x term is -22x. Take half its coefficient (-11). Square it (121) and add it to both sides. Add '121' to each side of the equation. -22x + 121 + x2 = -120 + 121 Reorder the terms: 121 + -22x + x2 = -120 + 121 Combine like terms: -120 + 121 = 1 121 + -22x + x2 = 1 Factor a perfect square on the left side: (x + -11)(x + -11) = 1 Calculate the square root of the right side: 1 Break this problem into two subproblems by setting (x + -11) equal to 1 and -1.Subproblem 1
x + -11 = 1 Simplifying x + -11 = 1 Reorder the terms: -11 + x = 1 Solving -11 + x = 1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + x = 1 + 11 Combine like terms: -11 + 11 = 0 0 + x = 1 + 11 x = 1 + 11 Combine like terms: 1 + 11 = 12 x = 12 Simplifying x = 12Subproblem 2
x + -11 = -1 Simplifying x + -11 = -1 Reorder the terms: -11 + x = -1 Solving -11 + x = -1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + x = -1 + 11 Combine like terms: -11 + 11 = 0 0 + x = -1 + 11 x = -1 + 11 Combine like terms: -1 + 11 = 10 x = 10 Simplifying x = 10Solution
The solution to the problem is based on the solutions from the subproblems. x = {12, 10}
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