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Simplifying 1174.25 = (32 + x)(24 + x) Multiply (32 + x) * (24 + x) 1174.25 = (32(24 + x) + x(24 + x)) 1174.25 = ((24 * 32 + x * 32) + x(24 + x)) 1174.25 = ((768 + 32x) + x(24 + x)) 1174.25 = (768 + 32x + (24 * x + x * x)) 1174.25 = (768 + 32x + (24x + x2)) Combine like terms: 32x + 24x = 56x 1174.25 = (768 + 56x + x2) Solving 1174.25 = 768 + 56x + x2 Solving for variable 'x'. Combine like terms: 1174.25 + -768 = 406.25 406.25 + -56x + -1x2 = 768 + 56x + x2 + -768 + -56x + -1x2 Reorder the terms: 406.25 + -56x + -1x2 = 768 + -768 + 56x + -56x + x2 + -1x2 Combine like terms: 768 + -768 = 0 406.25 + -56x + -1x2 = 0 + 56x + -56x + x2 + -1x2 406.25 + -56x + -1x2 = 56x + -56x + x2 + -1x2 Combine like terms: 56x + -56x = 0 406.25 + -56x + -1x2 = 0 + x2 + -1x2 406.25 + -56x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 406.25 + -56x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -406.25 + 56x + x2 = 0 Move the constant term to the right: Add '406.25' to each side of the equation. -406.25 + 56x + 406.25 + x2 = 0 + 406.25 Reorder the terms: -406.25 + 406.25 + 56x + x2 = 0 + 406.25 Combine like terms: -406.25 + 406.25 = 0.00 0.00 + 56x + x2 = 0 + 406.25 56x + x2 = 0 + 406.25 Combine like terms: 0 + 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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