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Simplifying x = 0.0216 + 0.42x + x2 Solving x = 0.0216 + 0.42x + x2 Solving for variable 'x'. Reorder the terms: -0.0216 + x + -0.42x + -1x2 = 0.0216 + 0.42x + x2 + -0.0216 + -0.42x + -1x2 Combine like terms: x + -0.42x = 0.58x -0.0216 + 0.58x + -1x2 = 0.0216 + 0.42x + x2 + -0.0216 + -0.42x + -1x2 Reorder the terms: -0.0216 + 0.58x + -1x2 = 0.0216 + -0.0216 + 0.42x + -0.42x + x2 + -1x2 Combine like terms: 0.0216 + -0.0216 = 0.0000 -0.0216 + 0.58x + -1x2 = 0.0000 + 0.42x + -0.42x + x2 + -1x2 -0.0216 + 0.58x + -1x2 = 0.42x + -0.42x + x2 + -1x2 Combine like terms: 0.42x + -0.42x = 0.00 -0.0216 + 0.58x + -1x2 = 0.00 + x2 + -1x2 -0.0216 + 0.58x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -0.0216 + 0.58x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 0.0216 + -0.58x + x2 = 0 Move the constant term to the right: Add '-0.0216' to each side of the equation. 0.0216 + -0.58x + -0.0216 + x2 = 0 + -0.0216 Reorder the terms: 0.0216 + -0.0216 + -0.58x + x2 = 0 + -0.0216 Combine like terms: 0.0216 + -0.0216 = 0.0000 0.0000 + -0.58x + x2 = 0 + -0.0216 -0.58x + x2 = 0 + -0.0216 Combine like terms: 0 + -0.0216 = -0.0216 -0.58x + x2 = -0.0216 The x term is -0.58x. Take half its coefficient (-0.29). Square it (0.0841) and add it to both sides. Add '0.0841' to each side of the equation. -0.58x + 0.0841 + x2 = -0.0216 + 0.0841 Reorder the terms: 0.0841 + -0.58x + x2 = -0.0216 + 0.0841 Combine like terms: -0.0216 + 0.0841 = 0.0625 0.0841 + -0.58x + x2 = 0.0625 Factor a perfect square on the left side: (x + -0.29)(x + -0.29) = 0.0625 Calculate the square root of the right side: 0.25 Break this problem into two subproblems by setting (x + -0.29) equal to 0.25 and -0.25.Subproblem 1
x + -0.29 = 0.25 Simplifying x + -0.29 = 0.25 Reorder the terms: -0.29 + x = 0.25 Solving -0.29 + x = 0.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.29' to each side of the equation. -0.29 + 0.29 + x = 0.25 + 0.29 Combine like terms: -0.29 + 0.29 = 0.00 0.00 + x = 0.25 + 0.29 x = 0.25 + 0.29 Combine like terms: 0.25 + 0.29 = 0.54 x = 0.54 Simplifying x = 0.54Subproblem 2
x + -0.29 = -0.25 Simplifying x + -0.29 = -0.25 Reorder the terms: -0.29 + x = -0.25 Solving -0.29 + x = -0.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.29' to each side of the equation. -0.29 + 0.29 + x = -0.25 + 0.29 Combine like terms: -0.29 + 0.29 = 0.00 0.00 + x = -0.25 + 0.29 x = -0.25 + 0.29 Combine like terms: -0.25 + 0.29 = 0.04 x = 0.04 Simplifying x = 0.04Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.54, 0.04}
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