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Simplifying (0.5x + 0.2)(x + 0.7) = 0 Reorder the terms: (0.2 + 0.5x)(x + 0.7) = 0 Reorder the terms: (0.2 + 0.5x)(0.7 + x) = 0 Multiply (0.2 + 0.5x) * (0.7 + x) (0.2(0.7 + x) + 0.5x * (0.7 + x)) = 0 ((0.7 * 0.2 + x * 0.2) + 0.5x * (0.7 + x)) = 0 ((0.14 + 0.2x) + 0.5x * (0.7 + x)) = 0 (0.14 + 0.2x + (0.7 * 0.5x + x * 0.5x)) = 0 (0.14 + 0.2x + (0.35x + 0.5x2)) = 0 Combine like terms: 0.2x + 0.35x = 0.55x (0.14 + 0.55x + 0.5x2) = 0 Solving 0.14 + 0.55x + 0.5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 0.28 + 1.1x + x2 = 0 Move the constant term to the right: Add '-0.28' to each side of the equation. 0.28 + 1.1x + -0.28 + x2 = 0 + -0.28 Reorder the terms: 0.28 + -0.28 + 1.1x + x2 = 0 + -0.28 Combine like terms: 0.28 + -0.28 = 0.00 0.00 + 1.1x + x2 = 0 + -0.28 1.1x + x2 = 0 + -0.28 Combine like terms: 0 + -0.28 = -0.28 1.1x + x2 = -0.28 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 = -0.28 + 0.3025 Reorder the terms: 0.3025 + 1.1x + x2 = -0.28 + 0.3025 Combine like terms: -0.28 + 0.3025 = 0.0225 0.3025 + 1.1x + x2 = 0.0225 Factor a perfect square on the left side: (x + 0.55)(x + 0.55) = 0.0225 Calculate the square root of the right side: 0.15 Break this problem into two subproblems by setting (x + 0.55) equal to 0.15 and -0.15.Subproblem 1
x + 0.55 = 0.15 Simplifying x + 0.55 = 0.15 Reorder the terms: 0.55 + x = 0.15 Solving 0.55 + x = 0.15 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 = 0.15 + -0.55 Combine like terms: 0.55 + -0.55 = 0.00 0.00 + x = 0.15 + -0.55 x = 0.15 + -0.55 Combine like terms: 0.15 + -0.55 = -0.4 x = -0.4 Simplifying x = -0.4Subproblem 2
x + 0.55 = -0.15 Simplifying x + 0.55 = -0.15 Reorder the terms: 0.55 + x = -0.15 Solving 0.55 + x = -0.15 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 = -0.15 + -0.55 Combine like terms: 0.55 + -0.55 = 0.00 0.00 + x = -0.15 + -0.55 x = -0.15 + -0.55 Combine like terms: -0.15 + -0.55 = -0.7 x = -0.7 Simplifying x = -0.7Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.4, -0.7}
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