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Simplifying (24 + x)(32 + x) = 1174.25 Multiply (24 + x) * (32 + x) (24(32 + x) + x(32 + x)) = 1174.25 ((32 * 24 + x * 24) + x(32 + x)) = 1174.25 ((768 + 24x) + x(32 + x)) = 1174.25 (768 + 24x + (32 * x + x * x)) = 1174.25 (768 + 24x + (32x + x2)) = 1174.25 Combine like terms: 24x + 32x = 56x (768 + 56x + x2) = 1174.25 Solving 768 + 56x + x2 = 1174.25 Solving for variable 'x'. Reorder the terms: 768 + -1174.25 + 56x + x2 = 1174.25 + -1174.25 Combine like terms: 768 + -1174.25 = -406.25 -406.25 + 56x + x2 = 1174.25 + -1174.25 Combine like terms: 1174.25 + -1174.25 = 0.00 -406.25 + 56x + x2 = 0.00 Begin completing the square. Move the constant term to the right: Add '406.25' to each side of the equation. -406.25 + 56x + 406.25 + x2 = 0.00 + 406.25 Reorder the terms: -406.25 + 406.25 + 56x + x2 = 0.00 + 406.25 Combine like terms: -406.25 + 406.25 = 0.00 0.00 + 56x + x2 = 0.00 + 406.25 56x + x2 = 0.00 + 406.25 Combine like terms: 0.00 + 406.25 = 406.25 56x + x2 = 406.25 The x term is 56x. Take half its coefficient (28). Square it (784) and add it to both sides. Add '784' to each side of the equation. 56x + 784 + x2 = 406.25 + 784 Reorder the terms: 784 + 56x + x2 = 406.25 + 784 Combine like terms: 406.25 + 784 = 1190.25 784 + 56x + x2 = 1190.25 Factor a perfect square on the left side: (x + 28)(x + 28) = 1190.25 Calculate the square root of the right side: 34.5 Break this problem into two subproblems by setting (x + 28) equal to 34.5 and -34.5.Subproblem 1
x + 28 = 34.5 Simplifying x + 28 = 34.5 Reorder the terms: 28 + x = 34.5 Solving 28 + x = 34.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-28' to each side of the equation. 28 + -28 + x = 34.5 + -28 Combine like terms: 28 + -28 = 0 0 + x = 34.5 + -28 x = 34.5 + -28 Combine like terms: 34.5 + -28 = 6.5 x = 6.5 Simplifying x = 6.5Subproblem 2
x + 28 = -34.5 Simplifying x + 28 = -34.5 Reorder the terms: 28 + x = -34.5 Solving 28 + x = -34.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-28' to each side of the equation. 28 + -28 + x = -34.5 + -28 Combine like terms: 28 + -28 = 0 0 + x = -34.5 + -28 x = -34.5 + -28 Combine like terms: -34.5 + -28 = -62.5 x = -62.5 Simplifying x = -62.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.5, -62.5}
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