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Simplifying 90 + x(2x + 15) = 180 Reorder the terms: 90 + x(15 + 2x) = 180 90 + (15 * x + 2x * x) = 180 90 + (15x + 2x2) = 180 Solving 90 + 15x + 2x2 = 180 Solving for variable 'x'. Reorder the terms: 90 + -180 + 15x + 2x2 = 180 + -180 Combine like terms: 90 + -180 = -90 -90 + 15x + 2x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -90 + 15x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -45 + 7.5x + x2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + 7.5x + 45 + x2 = 0 + 45 Reorder the terms: -45 + 45 + 7.5x + x2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 7.5x + x2 = 0 + 45 7.5x + x2 = 0 + 45 Combine like terms: 0 + 45 = 45 7.5x + x2 = 45 The x term is 7.5x. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5x + 14.0625 + x2 = 45 + 14.0625 Reorder the terms: 14.0625 + 7.5x + x2 = 45 + 14.0625 Combine like terms: 45 + 14.0625 = 59.0625 14.0625 + 7.5x + x2 = 59.0625 Factor a perfect square on the left side: (x + 3.75)(x + 3.75) = 59.0625 Calculate the square root of the right side: 7.685213074 Break this problem into two subproblems by setting (x + 3.75) equal to 7.685213074 and -7.685213074.Subproblem 1
x + 3.75 = 7.685213074 Simplifying x + 3.75 = 7.685213074 Reorder the terms: 3.75 + x = 7.685213074 Solving 3.75 + x = 7.685213074 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + x = 7.685213074 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = 7.685213074 + -3.75 x = 7.685213074 + -3.75 Combine like terms: 7.685213074 + -3.75 = 3.935213074 x = 3.935213074 Simplifying x = 3.935213074Subproblem 2
x + 3.75 = -7.685213074 Simplifying x + 3.75 = -7.685213074 Reorder the terms: 3.75 + x = -7.685213074 Solving 3.75 + x = -7.685213074 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + x = -7.685213074 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = -7.685213074 + -3.75 x = -7.685213074 + -3.75 Combine like terms: -7.685213074 + -3.75 = -11.435213074 x = -11.435213074 Simplifying x = -11.435213074Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.935213074, -11.435213074}
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