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Simplifying 5 = -0.024x2 + 0.53x + 3 Reorder the terms: 5 = 3 + 0.53x + -0.024x2 Solving 5 = 3 + 0.53x + -0.024x2 Solving for variable 'x'. Combine like terms: 5 + -3 = 2 2 + -0.53x + 0.024x2 = 3 + 0.53x + -0.024x2 + -3 + -0.53x + 0.024x2 Reorder the terms: 2 + -0.53x + 0.024x2 = 3 + -3 + 0.53x + -0.53x + -0.024x2 + 0.024x2 Combine like terms: 3 + -3 = 0 2 + -0.53x + 0.024x2 = 0 + 0.53x + -0.53x + -0.024x2 + 0.024x2 2 + -0.53x + 0.024x2 = 0.53x + -0.53x + -0.024x2 + 0.024x2 Combine like terms: 0.53x + -0.53x = 0.00 2 + -0.53x + 0.024x2 = 0.00 + -0.024x2 + 0.024x2 2 + -0.53x + 0.024x2 = -0.024x2 + 0.024x2 Combine like terms: -0.024x2 + 0.024x2 = 0.000 2 + -0.53x + 0.024x2 = 0.000 Begin completing the square. Divide all terms by 0.024 the coefficient of the squared term: Divide each side by '0.024'. 83.33333333 + -22.08333333x + x2 = 0 Move the constant term to the right: Add '-83.33333333' to each side of the equation. 83.33333333 + -22.08333333x + -83.33333333 + x2 = 0 + -83.33333333 Reorder the terms: 83.33333333 + -83.33333333 + -22.08333333x + x2 = 0 + -83.33333333 Combine like terms: 83.33333333 + -83.33333333 = 0.00000000 0.00000000 + -22.08333333x + x2 = 0 + -83.33333333 -22.08333333x + x2 = 0 + -83.33333333 Combine like terms: 0 + -83.33333333 = -83.33333333 -22.08333333x + x2 = -83.33333333 The x term is -22.08333333x. Take half its coefficient (-11.04166667). Square it (121.9184029) and add it to both sides. Add '121.9184029' to each side of the equation. -22.08333333x + 121.9184029 + x2 = -83.33333333 + 121.9184029 Reorder the terms: 121.9184029 + -22.08333333x + x2 = -83.33333333 + 121.9184029 Combine like terms: -83.33333333 + 121.9184029 = 38.58506957 121.9184029 + -22.08333333x + x2 = 38.58506957 Factor a perfect square on the left side: (x + -11.04166667)(x + -11.04166667) = 38.58506957 Calculate the square root of the right side: 6.211688142 Break this problem into two subproblems by setting (x + -11.04166667) equal to 6.211688142 and -6.211688142.Subproblem 1
x + -11.04166667 = 6.211688142 Simplifying x + -11.04166667 = 6.211688142 Reorder the terms: -11.04166667 + x = 6.211688142 Solving -11.04166667 + x = 6.211688142 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.04166667' to each side of the equation. -11.04166667 + 11.04166667 + x = 6.211688142 + 11.04166667 Combine like terms: -11.04166667 + 11.04166667 = 0.00000000 0.00000000 + x = 6.211688142 + 11.04166667 x = 6.211688142 + 11.04166667 Combine like terms: 6.211688142 + 11.04166667 = 17.253354812 x = 17.253354812 Simplifying x = 17.253354812Subproblem 2
x + -11.04166667 = -6.211688142 Simplifying x + -11.04166667 = -6.211688142 Reorder the terms: -11.04166667 + x = -6.211688142 Solving -11.04166667 + x = -6.211688142 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.04166667' to each side of the equation. -11.04166667 + 11.04166667 + x = -6.211688142 + 11.04166667 Combine like terms: -11.04166667 + 11.04166667 = 0.00000000 0.00000000 + x = -6.211688142 + 11.04166667 x = -6.211688142 + 11.04166667 Combine like terms: -6.211688142 + 11.04166667 = 4.829978528 x = 4.829978528 Simplifying x = 4.829978528Solution
The solution to the problem is based on the solutions from the subproblems. x = {17.253354812, 4.829978528}
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