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Simplifying (3 + x)(7 + x) = 42 Multiply (3 + x) * (7 + x) (3(7 + x) + x(7 + x)) = 42 ((7 * 3 + x * 3) + x(7 + x)) = 42 ((21 + 3x) + x(7 + x)) = 42 (21 + 3x + (7 * x + x * x)) = 42 (21 + 3x + (7x + x2)) = 42 Combine like terms: 3x + 7x = 10x (21 + 10x + x2) = 42 Solving 21 + 10x + x2 = 42 Solving for variable 'x'. Reorder the terms: 21 + -42 + 10x + x2 = 42 + -42 Combine like terms: 21 + -42 = -21 -21 + 10x + x2 = 42 + -42 Combine like terms: 42 + -42 = 0 -21 + 10x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '21' to each side of the equation. -21 + 10x + 21 + x2 = 0 + 21 Reorder the terms: -21 + 21 + 10x + x2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 10x + x2 = 0 + 21 10x + x2 = 0 + 21 Combine like terms: 0 + 21 = 21 10x + x2 = 21 The x term is 10x. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10x + 25 + x2 = 21 + 25 Reorder the terms: 25 + 10x + x2 = 21 + 25 Combine like terms: 21 + 25 = 46 25 + 10x + x2 = 46 Factor a perfect square on the left side: (x + 5)(x + 5) = 46 Calculate the square root of the right side: 6.782329983 Break this problem into two subproblems by setting (x + 5) equal to 6.782329983 and -6.782329983.Subproblem 1
x + 5 = 6.782329983 Simplifying x + 5 = 6.782329983 Reorder the terms: 5 + x = 6.782329983 Solving 5 + x = 6.782329983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = 6.782329983 + -5 Combine like terms: 5 + -5 = 0 0 + x = 6.782329983 + -5 x = 6.782329983 + -5 Combine like terms: 6.782329983 + -5 = 1.782329983 x = 1.782329983 Simplifying x = 1.782329983Subproblem 2
x + 5 = -6.782329983 Simplifying x + 5 = -6.782329983 Reorder the terms: 5 + x = -6.782329983 Solving 5 + x = -6.782329983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = -6.782329983 + -5 Combine like terms: 5 + -5 = 0 0 + x = -6.782329983 + -5 x = -6.782329983 + -5 Combine like terms: -6.782329983 + -5 = -11.782329983 x = -11.782329983 Simplifying x = -11.782329983Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.782329983, -11.782329983}
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