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Simplifying (0.4x + 0.2)(x + 0.6) = 0 Reorder the terms: (0.2 + 0.4x)(x + 0.6) = 0 Reorder the terms: (0.2 + 0.4x)(0.6 + x) = 0 Multiply (0.2 + 0.4x) * (0.6 + x) (0.2(0.6 + x) + 0.4x * (0.6 + x)) = 0 ((0.6 * 0.2 + x * 0.2) + 0.4x * (0.6 + x)) = 0 ((0.12 + 0.2x) + 0.4x * (0.6 + x)) = 0 (0.12 + 0.2x + (0.6 * 0.4x + x * 0.4x)) = 0 (0.12 + 0.2x + (0.24x + 0.4x2)) = 0 Combine like terms: 0.2x + 0.24x = 0.44x (0.12 + 0.44x + 0.4x2) = 0 Solving 0.12 + 0.44x + 0.4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.4 the coefficient of the squared term: Divide each side by '0.4'. 0.3 + 1.1x + x2 = 0 Move the constant term to the right: Add '-0.3' to each side of the equation. 0.3 + 1.1x + -0.3 + x2 = 0 + -0.3 Reorder the terms: 0.3 + -0.3 + 1.1x + x2 = 0 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + 1.1x + x2 = 0 + -0.3 1.1x + x2 = 0 + -0.3 Combine like terms: 0 + -0.3 = -0.3 1.1x + x2 = -0.3 The x term is 1.1x. Take half its coefficient (0.55). Square it (0.3025) and add it to both sides. Add '0.3025' to each side of the equation. 1.1x + 0.3025 + x2 = -0.3 + 0.3025 Reorder the terms: 0.3025 + 1.1x + x2 = -0.3 + 0.3025 Combine like terms: -0.3 + 0.3025 = 0.0025 0.3025 + 1.1x + x2 = 0.0025 Factor a perfect square on the left side: (x + 0.55)(x + 0.55) = 0.0025 Calculate the square root of the right side: 0.05 Break this problem into two subproblems by setting (x + 0.55) equal to 0.05 and -0.05.Subproblem 1
x + 0.55 = 0.05 Simplifying x + 0.55 = 0.05 Reorder the terms: 0.55 + x = 0.05 Solving 0.55 + x = 0.05 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.55' to each side of the equation. 0.55 + -0.55 + x = 0.05 + -0.55 Combine like terms: 0.55 + -0.55 = 0.00 0.00 + x = 0.05 + -0.55 x = 0.05 + -0.55 Combine like terms: 0.05 + -0.55 = -0.5 x = -0.5 Simplifying x = -0.5Subproblem 2
x + 0.55 = -0.05 Simplifying x + 0.55 = -0.05 Reorder the terms: 0.55 + x = -0.05 Solving 0.55 + x = -0.05 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.55' to each side of the equation. 0.55 + -0.55 + x = -0.05 + -0.55 Combine like terms: 0.55 + -0.55 = 0.00 0.00 + x = -0.05 + -0.55 x = -0.05 + -0.55 Combine like terms: -0.05 + -0.55 = -0.6 x = -0.6 Simplifying x = -0.6Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.5, -0.6}
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