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Simplifying (2x + 9)(x) = 460 Reorder the terms: (9 + 2x)(x) = 460 Reorder the terms for easier multiplication: x(9 + 2x) = 460 (9 * x + 2x * x) = 460 (9x + 2x2) = 460 Solving 9x + 2x2 = 460 Solving for variable 'x'. Reorder the terms: -460 + 9x + 2x2 = 460 + -460 Combine like terms: 460 + -460 = 0 -460 + 9x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -230 + 4.5x + x2 = 0 Move the constant term to the right: Add '230' to each side of the equation. -230 + 4.5x + 230 + x2 = 0 + 230 Reorder the terms: -230 + 230 + 4.5x + x2 = 0 + 230 Combine like terms: -230 + 230 = 0 0 + 4.5x + x2 = 0 + 230 4.5x + x2 = 0 + 230 Combine like terms: 0 + 230 = 230 4.5x + x2 = 230 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 230 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 230 + 5.0625 Combine like terms: 230 + 5.0625 = 235.0625 5.0625 + 4.5x + x2 = 235.0625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 235.0625 Calculate the square root of the right side: 15.331748106 Break this problem into two subproblems by setting (x + 2.25) equal to 15.331748106 and -15.331748106.Subproblem 1
x + 2.25 = 15.331748106 Simplifying x + 2.25 = 15.331748106 Reorder the terms: 2.25 + x = 15.331748106 Solving 2.25 + x = 15.331748106 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 15.331748106 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 15.331748106 + -2.25 x = 15.331748106 + -2.25 Combine like terms: 15.331748106 + -2.25 = 13.081748106 x = 13.081748106 Simplifying x = 13.081748106Subproblem 2
x + 2.25 = -15.331748106 Simplifying x + 2.25 = -15.331748106 Reorder the terms: 2.25 + x = -15.331748106 Solving 2.25 + x = -15.331748106 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -15.331748106 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -15.331748106 + -2.25 x = -15.331748106 + -2.25 Combine like terms: -15.331748106 + -2.25 = -17.581748106 x = -17.581748106 Simplifying x = -17.581748106Solution
The solution to the problem is based on the solutions from the subproblems. x = {13.081748106, -17.581748106}
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