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Simplifying 3x(22 + -2x) = 5x + 32 (22 * 3x + -2x * 3x) = 5x + 32 (66x + -6x2) = 5x + 32 Reorder the terms: 66x + -6x2 = 32 + 5x Solving 66x + -6x2 = 32 + 5x Solving for variable 'x'. Reorder the terms: -32 + 66x + -5x + -6x2 = 32 + 5x + -32 + -5x Combine like terms: 66x + -5x = 61x -32 + 61x + -6x2 = 32 + 5x + -32 + -5x Reorder the terms: -32 + 61x + -6x2 = 32 + -32 + 5x + -5x Combine like terms: 32 + -32 = 0 -32 + 61x + -6x2 = 0 + 5x + -5x -32 + 61x + -6x2 = 5x + -5x Combine like terms: 5x + -5x = 0 -32 + 61x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. 5.333333333 + -10.16666667x + x2 = 0 Move the constant term to the right: Add '-5.333333333' to each side of the equation. 5.333333333 + -10.16666667x + -5.333333333 + x2 = 0 + -5.333333333 Reorder the terms: 5.333333333 + -5.333333333 + -10.16666667x + x2 = 0 + -5.333333333 Combine like terms: 5.333333333 + -5.333333333 = 0.000000000 0.000000000 + -10.16666667x + x2 = 0 + -5.333333333 -10.16666667x + x2 = 0 + -5.333333333 Combine like terms: 0 + -5.333333333 = -5.333333333 -10.16666667x + x2 = -5.333333333 The x term is -10.16666667x. Take half its coefficient (-5.083333335). Square it (25.84027779) and add it to both sides. Add '25.84027779' to each side of the equation. -10.16666667x + 25.84027779 + x2 = -5.333333333 + 25.84027779 Reorder the terms: 25.84027779 + -10.16666667x + x2 = -5.333333333 + 25.84027779 Combine like terms: -5.333333333 + 25.84027779 = 20.506944457 25.84027779 + -10.16666667x + x2 = 20.506944457 Factor a perfect square on the left side: (x + -5.083333335)(x + -5.083333335) = 20.506944457 Calculate the square root of the right side: 4.528459391 Break this problem into two subproblems by setting (x + -5.083333335) equal to 4.528459391 and -4.528459391.Subproblem 1
x + -5.083333335 = 4.528459391 Simplifying x + -5.083333335 = 4.528459391 Reorder the terms: -5.083333335 + x = 4.528459391 Solving -5.083333335 + x = 4.528459391 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.083333335' to each side of the equation. -5.083333335 + 5.083333335 + x = 4.528459391 + 5.083333335 Combine like terms: -5.083333335 + 5.083333335 = 0.000000000 0.000000000 + x = 4.528459391 + 5.083333335 x = 4.528459391 + 5.083333335 Combine like terms: 4.528459391 + 5.083333335 = 9.611792726 x = 9.611792726 Simplifying x = 9.611792726Subproblem 2
x + -5.083333335 = -4.528459391 Simplifying x + -5.083333335 = -4.528459391 Reorder the terms: -5.083333335 + x = -4.528459391 Solving -5.083333335 + x = -4.528459391 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.083333335' to each side of the equation. -5.083333335 + 5.083333335 + x = -4.528459391 + 5.083333335 Combine like terms: -5.083333335 + 5.083333335 = 0.000000000 0.000000000 + x = -4.528459391 + 5.083333335 x = -4.528459391 + 5.083333335 Combine like terms: -4.528459391 + 5.083333335 = 0.554873944 x = 0.554873944 Simplifying x = 0.554873944Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.611792726, 0.554873944}
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