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Simplifying (4 + x)(3 + x) = 1000 Multiply (4 + x) * (3 + x) (4(3 + x) + x(3 + x)) = 1000 ((3 * 4 + x * 4) + x(3 + x)) = 1000 ((12 + 4x) + x(3 + x)) = 1000 (12 + 4x + (3 * x + x * x)) = 1000 (12 + 4x + (3x + x2)) = 1000 Combine like terms: 4x + 3x = 7x (12 + 7x + x2) = 1000 Solving 12 + 7x + x2 = 1000 Solving for variable 'x'. Reorder the terms: 12 + -1000 + 7x + x2 = 1000 + -1000 Combine like terms: 12 + -1000 = -988 -988 + 7x + x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -988 + 7x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '988' to each side of the equation. -988 + 7x + 988 + x2 = 0 + 988 Reorder the terms: -988 + 988 + 7x + x2 = 0 + 988 Combine like terms: -988 + 988 = 0 0 + 7x + x2 = 0 + 988 7x + x2 = 0 + 988 Combine like terms: 0 + 988 = 988 7x + x2 = 988 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 988 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 988 + 12.25 Combine like terms: 988 + 12.25 = 1000.25 12.25 + 7x + x2 = 1000.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 1000.25 Calculate the square root of the right side: 31.626729202 Break this problem into two subproblems by setting (x + 3.5) equal to 31.626729202 and -31.626729202.Subproblem 1
x + 3.5 = 31.626729202 Simplifying x + 3.5 = 31.626729202 Reorder the terms: 3.5 + x = 31.626729202 Solving 3.5 + x = 31.626729202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 31.626729202 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 31.626729202 + -3.5 x = 31.626729202 + -3.5 Combine like terms: 31.626729202 + -3.5 = 28.126729202 x = 28.126729202 Simplifying x = 28.126729202Subproblem 2
x + 3.5 = -31.626729202 Simplifying x + 3.5 = -31.626729202 Reorder the terms: 3.5 + x = -31.626729202 Solving 3.5 + x = -31.626729202 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -31.626729202 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -31.626729202 + -3.5 x = -31.626729202 + -3.5 Combine like terms: -31.626729202 + -3.5 = -35.126729202 x = -35.126729202 Simplifying x = -35.126729202Solution
The solution to the problem is based on the solutions from the subproblems. x = {28.126729202, -35.126729202}
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