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Simplifying (5 + x)(10 + x) = 54 Multiply (5 + x) * (10 + x) (5(10 + x) + x(10 + x)) = 54 ((10 * 5 + x * 5) + x(10 + x)) = 54 ((50 + 5x) + x(10 + x)) = 54 (50 + 5x + (10 * x + x * x)) = 54 (50 + 5x + (10x + x2)) = 54 Combine like terms: 5x + 10x = 15x (50 + 15x + x2) = 54 Solving 50 + 15x + x2 = 54 Solving for variable 'x'. Reorder the terms: 50 + -54 + 15x + x2 = 54 + -54 Combine like terms: 50 + -54 = -4 -4 + 15x + x2 = 54 + -54 Combine like terms: 54 + -54 = 0 -4 + 15x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 15x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + 15x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 15x + x2 = 0 + 4 15x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 15x + x2 = 4 The x term is 15x. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15x + 56.25 + x2 = 4 + 56.25 Reorder the terms: 56.25 + 15x + x2 = 4 + 56.25 Combine like terms: 4 + 56.25 = 60.25 56.25 + 15x + x2 = 60.25 Factor a perfect square on the left side: (x + 7.5)(x + 7.5) = 60.25 Calculate the square root of the right side: 7.762087348 Break this problem into two subproblems by setting (x + 7.5) equal to 7.762087348 and -7.762087348.Subproblem 1
x + 7.5 = 7.762087348 Simplifying x + 7.5 = 7.762087348 Reorder the terms: 7.5 + x = 7.762087348 Solving 7.5 + x = 7.762087348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = 7.762087348 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 7.762087348 + -7.5 x = 7.762087348 + -7.5 Combine like terms: 7.762087348 + -7.5 = 0.262087348 x = 0.262087348 Simplifying x = 0.262087348Subproblem 2
x + 7.5 = -7.762087348 Simplifying x + 7.5 = -7.762087348 Reorder the terms: 7.5 + x = -7.762087348 Solving 7.5 + x = -7.762087348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = -7.762087348 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -7.762087348 + -7.5 x = -7.762087348 + -7.5 Combine like terms: -7.762087348 + -7.5 = -15.262087348 x = -15.262087348 Simplifying x = -15.262087348Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.262087348, -15.262087348}
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