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Simplifying 2x(x + 10) + x = 180 Reorder the terms: 2x(10 + x) + x = 180 (10 * 2x + x * 2x) + x = 180 (20x + 2x2) + x = 180 Reorder the terms: 20x + x + 2x2 = 180 Combine like terms: 20x + x = 21x 21x + 2x2 = 180 Solving 21x + 2x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + 21x + 2x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 21x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -90 + 10.5x + x2 = 0 Move the constant term to the right: Add '90' to each side of the equation. -90 + 10.5x + 90 + x2 = 0 + 90 Reorder the terms: -90 + 90 + 10.5x + x2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 10.5x + x2 = 0 + 90 10.5x + x2 = 0 + 90 Combine like terms: 0 + 90 = 90 10.5x + x2 = 90 The x term is 10.5x. Take half its coefficient (5.25). Square it (27.5625) and add it to both sides. Add '27.5625' to each side of the equation. 10.5x + 27.5625 + x2 = 90 + 27.5625 Reorder the terms: 27.5625 + 10.5x + x2 = 90 + 27.5625 Combine like terms: 90 + 27.5625 = 117.5625 27.5625 + 10.5x + x2 = 117.5625 Factor a perfect square on the left side: (x + 5.25)(x + 5.25) = 117.5625 Calculate the square root of the right side: 10.842624221 Break this problem into two subproblems by setting (x + 5.25) equal to 10.842624221 and -10.842624221.Subproblem 1
x + 5.25 = 10.842624221 Simplifying x + 5.25 = 10.842624221 Reorder the terms: 5.25 + x = 10.842624221 Solving 5.25 + x = 10.842624221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + x = 10.842624221 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + x = 10.842624221 + -5.25 x = 10.842624221 + -5.25 Combine like terms: 10.842624221 + -5.25 = 5.592624221 x = 5.592624221 Simplifying x = 5.592624221Subproblem 2
x + 5.25 = -10.842624221 Simplifying x + 5.25 = -10.842624221 Reorder the terms: 5.25 + x = -10.842624221 Solving 5.25 + x = -10.842624221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + x = -10.842624221 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + x = -10.842624221 + -5.25 x = -10.842624221 + -5.25 Combine like terms: -10.842624221 + -5.25 = -16.092624221 x = -16.092624221 Simplifying x = -16.092624221Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.592624221, -16.092624221}
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