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Simplifying (0.3 + 2x)(0.5 + 2x) = 5.3018 Multiply (0.3 + 2x) * (0.5 + 2x) (0.3(0.5 + 2x) + 2x * (0.5 + 2x)) = 5.3018 ((0.5 * 0.3 + 2x * 0.3) + 2x * (0.5 + 2x)) = 5.3018 ((0.15 + 0.6x) + 2x * (0.5 + 2x)) = 5.3018 (0.15 + 0.6x + (0.5 * 2x + 2x * 2x)) = 5.3018 (0.15 + 0.6x + (1x + 4x2)) = 5.3018 Combine like terms: 0.6x + 1x = 1.6x (0.15 + 1.6x + 4x2) = 5.3018 Solving 0.15 + 1.6x + 4x2 = 5.3018 Solving for variable 'x'. Reorder the terms: 0.15 + -5.3018 + 1.6x + 4x2 = 5.3018 + -5.3018 Combine like terms: 0.15 + -5.3018 = -5.1518 -5.1518 + 1.6x + 4x2 = 5.3018 + -5.3018 Combine like terms: 5.3018 + -5.3018 = 0.0000 -5.1518 + 1.6x + 4x2 = 0.0000 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.28795 + 0.4x + x2 = 0.0000 Move the constant term to the right: Add '1.28795' to each side of the equation. -1.28795 + 0.4x + 1.28795 + x2 = 0.0000 + 1.28795 Reorder the terms: -1.28795 + 1.28795 + 0.4x + x2 = 0.0000 + 1.28795 Combine like terms: -1.28795 + 1.28795 = 0.00000 0.00000 + 0.4x + x2 = 0.0000 + 1.28795 0.4x + x2 = 0.0000 + 1.28795 Combine like terms: 0.0000 + 1.28795 = 1.28795 0.4x + x2 = 1.28795 The x term is 0.4x. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4x + 0.04 + x2 = 1.28795 + 0.04 Reorder the terms: 0.04 + 0.4x + x2 = 1.28795 + 0.04 Combine like terms: 1.28795 + 0.04 = 1.32795 0.04 + 0.4x + x2 = 1.32795 Factor a perfect square on the left side: (x + 0.2)(x + 0.2) = 1.32795 Calculate the square root of the right side: 1.152367129 Break this problem into two subproblems by setting (x + 0.2) equal to 1.152367129 and -1.152367129.Subproblem 1
x + 0.2 = 1.152367129 Simplifying x + 0.2 = 1.152367129 Reorder the terms: 0.2 + x = 1.152367129 Solving 0.2 + x = 1.152367129 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + x = 1.152367129 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + x = 1.152367129 + -0.2 x = 1.152367129 + -0.2 Combine like terms: 1.152367129 + -0.2 = 0.952367129 x = 0.952367129 Simplifying x = 0.952367129Subproblem 2
x + 0.2 = -1.152367129 Simplifying x + 0.2 = -1.152367129 Reorder the terms: 0.2 + x = -1.152367129 Solving 0.2 + x = -1.152367129 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + x = -1.152367129 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + x = -1.152367129 + -0.2 x = -1.152367129 + -0.2 Combine like terms: -1.152367129 + -0.2 = -1.352367129 x = -1.352367129 Simplifying x = -1.352367129Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.952367129, -1.352367129}
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