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Simplifying (x + 0.2)(x + 3.9) = 0 Reorder the terms: (0.2 + x)(x + 3.9) = 0 Reorder the terms: (0.2 + x)(3.9 + x) = 0 Multiply (0.2 + x) * (3.9 + x) (0.2(3.9 + x) + x(3.9 + x)) = 0 ((3.9 * 0.2 + x * 0.2) + x(3.9 + x)) = 0 ((0.78 + 0.2x) + x(3.9 + x)) = 0 (0.78 + 0.2x + (3.9 * x + x * x)) = 0 (0.78 + 0.2x + (3.9x + x2)) = 0 Combine like terms: 0.2x + 3.9x = 4.1x (0.78 + 4.1x + x2) = 0 Solving 0.78 + 4.1x + x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '-0.78' to each side of the equation. 0.78 + 4.1x + -0.78 + x2 = 0 + -0.78 Reorder the terms: 0.78 + -0.78 + 4.1x + x2 = 0 + -0.78 Combine like terms: 0.78 + -0.78 = 0.00 0.00 + 4.1x + x2 = 0 + -0.78 4.1x + x2 = 0 + -0.78 Combine like terms: 0 + -0.78 = -0.78 4.1x + x2 = -0.78 The x term is 4.1x. Take half its coefficient (2.05). Square it (4.2025) and add it to both sides. Add '4.2025' to each side of the equation. 4.1x + 4.2025 + x2 = -0.78 + 4.2025 Reorder the terms: 4.2025 + 4.1x + x2 = -0.78 + 4.2025 Combine like terms: -0.78 + 4.2025 = 3.4225 4.2025 + 4.1x + x2 = 3.4225 Factor a perfect square on the left side: (x + 2.05)(x + 2.05) = 3.4225 Calculate the square root of the right side: 1.85 Break this problem into two subproblems by setting (x + 2.05) equal to 1.85 and -1.85.Subproblem 1
x + 2.05 = 1.85 Simplifying x + 2.05 = 1.85 Reorder the terms: 2.05 + x = 1.85 Solving 2.05 + x = 1.85 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + x = 1.85 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + x = 1.85 + -2.05 x = 1.85 + -2.05 Combine like terms: 1.85 + -2.05 = -0.2 x = -0.2 Simplifying x = -0.2Subproblem 2
x + 2.05 = -1.85 Simplifying x + 2.05 = -1.85 Reorder the terms: 2.05 + x = -1.85 Solving 2.05 + x = -1.85 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.05' to each side of the equation. 2.05 + -2.05 + x = -1.85 + -2.05 Combine like terms: 2.05 + -2.05 = 0.00 0.00 + x = -1.85 + -2.05 x = -1.85 + -2.05 Combine like terms: -1.85 + -2.05 = -3.9 x = -3.9 Simplifying x = -3.9Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.2, -3.9}
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