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Simplifying x + 2x(2x + 15) = 180 Reorder the terms: x + 2x(15 + 2x) = 180 x + (15 * 2x + 2x * 2x) = 180 x + (30x + 4x2) = 180 Combine like terms: x + 30x = 31x 31x + 4x2 = 180 Solving 31x + 4x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + 31x + 4x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 31x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -45 + 7.75x + x2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + 7.75x + 45 + x2 = 0 + 45 Reorder the terms: -45 + 45 + 7.75x + x2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 7.75x + x2 = 0 + 45 7.75x + x2 = 0 + 45 Combine like terms: 0 + 45 = 45 7.75x + x2 = 45 The x term is 7.75x. Take half its coefficient (3.875). Square it (15.015625) and add it to both sides. Add '15.015625' to each side of the equation. 7.75x + 15.015625 + x2 = 45 + 15.015625 Reorder the terms: 15.015625 + 7.75x + x2 = 45 + 15.015625 Combine like terms: 45 + 15.015625 = 60.015625 15.015625 + 7.75x + x2 = 60.015625 Factor a perfect square on the left side: (x + 3.875)(x + 3.875) = 60.015625 Calculate the square root of the right side: 7.746975216 Break this problem into two subproblems by setting (x + 3.875) equal to 7.746975216 and -7.746975216.Subproblem 1
x + 3.875 = 7.746975216 Simplifying x + 3.875 = 7.746975216 Reorder the terms: 3.875 + x = 7.746975216 Solving 3.875 + x = 7.746975216 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = 7.746975216 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = 7.746975216 + -3.875 x = 7.746975216 + -3.875 Combine like terms: 7.746975216 + -3.875 = 3.871975216 x = 3.871975216 Simplifying x = 3.871975216Subproblem 2
x + 3.875 = -7.746975216 Simplifying x + 3.875 = -7.746975216 Reorder the terms: 3.875 + x = -7.746975216 Solving 3.875 + x = -7.746975216 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = -7.746975216 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = -7.746975216 + -3.875 x = -7.746975216 + -3.875 Combine like terms: -7.746975216 + -3.875 = -11.621975216 x = -11.621975216 Simplifying x = -11.621975216Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.871975216, -11.621975216}
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