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Simplifying (0.9 + -2x)(0.9 + -2x) = 0 Multiply (0.9 + -2x) * (0.9 + -2x) (0.9(0.9 + -2x) + -2x * (0.9 + -2x)) = 0 ((0.9 * 0.9 + -2x * 0.9) + -2x * (0.9 + -2x)) = 0 ((0.81 + -1.8x) + -2x * (0.9 + -2x)) = 0 (0.81 + -1.8x + (0.9 * -2x + -2x * -2x)) = 0 (0.81 + -1.8x + (-1.8x + 4x2)) = 0 Combine like terms: -1.8x + -1.8x = -3.6x (0.81 + -3.6x + 4x2) = 0 Solving 0.81 + -3.6x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.2025 + -0.9x + x2 = 0 Move the constant term to the right: Add '-0.2025' to each side of the equation. 0.2025 + -0.9x + -0.2025 + x2 = 0 + -0.2025 Reorder the terms: 0.2025 + -0.2025 + -0.9x + x2 = 0 + -0.2025 Combine like terms: 0.2025 + -0.2025 = 0.0000 0.0000 + -0.9x + x2 = 0 + -0.2025 -0.9x + x2 = 0 + -0.2025 Combine like terms: 0 + -0.2025 = -0.2025 -0.9x + x2 = -0.2025 The x term is -0.9x. Take half its coefficient (-0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. -0.9x + 0.2025 + x2 = -0.2025 + 0.2025 Reorder the terms: 0.2025 + -0.9x + x2 = -0.2025 + 0.2025 Combine like terms: -0.2025 + 0.2025 = 0.0000 0.2025 + -0.9x + x2 = 0.0000 Factor a perfect square on the left side: (x + -0.45)(x + -0.45) = 0.0000 Calculate the square root of the right side: 0.0000 Break this problem into two subproblems by setting (x + -0.45) equal to 0.0000 and 0.0000.Subproblem 1
x + -0.45 = 0.0000 Simplifying x + -0.45 = 0.0000 Reorder the terms: -0.45 + x = 0.0000 Solving -0.45 + x = 0.0000 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + x = 0.0000 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + x = 0.0000 + 0.45 x = 0.0000 + 0.45 Combine like terms: 0.0000 + 0.45 = 0.45 x = 0.45 Simplifying x = 0.45Subproblem 2
x + -0.45 = 0.0000 Simplifying x + -0.45 = 0.0000 Reorder the terms: -0.45 + x = 0.0000 Solving -0.45 + x = 0.0000 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.45' to each side of the equation. -0.45 + 0.45 + x = 0.0000 + 0.45 Combine like terms: -0.45 + 0.45 = 0.00 0.00 + x = 0.0000 + 0.45 x = 0.0000 + 0.45 Combine like terms: 0.0000 + 0.45 = 0.45 x = 0.45 Simplifying x = 0.45Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.45, 0.45}
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