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Simplifying (x + 15)(x + 30) = 8000 Reorder the terms: (15 + x)(x + 30) = 8000 Reorder the terms: (15 + x)(30 + x) = 8000 Multiply (15 + x) * (30 + x) (15(30 + x) + x(30 + x)) = 8000 ((30 * 15 + x * 15) + x(30 + x)) = 8000 ((450 + 15x) + x(30 + x)) = 8000 (450 + 15x + (30 * x + x * x)) = 8000 (450 + 15x + (30x + x2)) = 8000 Combine like terms: 15x + 30x = 45x (450 + 45x + x2) = 8000 Solving 450 + 45x + x2 = 8000 Solving for variable 'x'. Reorder the terms: 450 + -8000 + 45x + x2 = 8000 + -8000 Combine like terms: 450 + -8000 = -7550 -7550 + 45x + x2 = 8000 + -8000 Combine like terms: 8000 + -8000 = 0 -7550 + 45x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '7550' to each side of the equation. -7550 + 45x + 7550 + x2 = 0 + 7550 Reorder the terms: -7550 + 7550 + 45x + x2 = 0 + 7550 Combine like terms: -7550 + 7550 = 0 0 + 45x + x2 = 0 + 7550 45x + x2 = 0 + 7550 Combine like terms: 0 + 7550 = 7550 45x + x2 = 7550 The x term is 45x. Take half its coefficient (22.5). Square it (506.25) and add it to both sides. Add '506.25' to each side of the equation. 45x + 506.25 + x2 = 7550 + 506.25 Reorder the terms: 506.25 + 45x + x2 = 7550 + 506.25 Combine like terms: 7550 + 506.25 = 8056.25 506.25 + 45x + x2 = 8056.25 Factor a perfect square on the left side: (x + 22.5)(x + 22.5) = 8056.25 Calculate the square root of the right side: 89.756615355 Break this problem into two subproblems by setting (x + 22.5) equal to 89.756615355 and -89.756615355.Subproblem 1
x + 22.5 = 89.756615355 Simplifying x + 22.5 = 89.756615355 Reorder the terms: 22.5 + x = 89.756615355 Solving 22.5 + x = 89.756615355 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + x = 89.756615355 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + x = 89.756615355 + -22.5 x = 89.756615355 + -22.5 Combine like terms: 89.756615355 + -22.5 = 67.256615355 x = 67.256615355 Simplifying x = 67.256615355Subproblem 2
x + 22.5 = -89.756615355 Simplifying x + 22.5 = -89.756615355 Reorder the terms: 22.5 + x = -89.756615355 Solving 22.5 + x = -89.756615355 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + x = -89.756615355 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + x = -89.756615355 + -22.5 x = -89.756615355 + -22.5 Combine like terms: -89.756615355 + -22.5 = -112.256615355 x = -112.256615355 Simplifying x = -112.256615355Solution
The solution to the problem is based on the solutions from the subproblems. x = {67.256615355, -112.256615355}
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