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Simplifying x(2x + 0.15) + (x + 0.05) = 1.8 Reorder the terms: x(0.15 + 2x) + (x + 0.05) = 1.8 (0.15 * x + 2x * x) + (x + 0.05) = 1.8 (0.15x + 2x2) + (x + 0.05) = 1.8 Reorder the terms: 0.15x + 2x2 + (0.05 + x) = 1.8 Remove parenthesis around (0.05 + x) 0.15x + 2x2 + 0.05 + x = 1.8 Reorder the terms: 0.05 + 0.15x + x + 2x2 = 1.8 Combine like terms: 0.15x + x = 1.15x 0.05 + 1.15x + 2x2 = 1.8 Solving 0.05 + 1.15x + 2x2 = 1.8 Solving for variable 'x'. Reorder the terms: 0.05 + -1.8 + 1.15x + 2x2 = 1.8 + -1.8 Combine like terms: 0.05 + -1.8 = -1.75 -1.75 + 1.15x + 2x2 = 1.8 + -1.8 Combine like terms: 1.8 + -1.8 = 0.0 -1.75 + 1.15x + 2x2 = 0.0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.875 + 0.575x + x2 = 0.0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + 0.575x + 0.875 + x2 = 0.0 + 0.875 Reorder the terms: -0.875 + 0.875 + 0.575x + x2 = 0.0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + 0.575x + x2 = 0.0 + 0.875 0.575x + x2 = 0.0 + 0.875 Combine like terms: 0.0 + 0.875 = 0.875 0.575x + x2 = 0.875 The x term is 0.575x. Take half its coefficient (0.2875). Square it (0.08265625) and add it to both sides. Add '0.08265625' to each side of the equation. 0.575x + 0.08265625 + x2 = 0.875 + 0.08265625 Reorder the terms: 0.08265625 + 0.575x + x2 = 0.875 + 0.08265625 Combine like terms: 0.875 + 0.08265625 = 0.95765625 0.08265625 + 0.575x + x2 = 0.95765625 Factor a perfect square on the left side: (x + 0.2875)(x + 0.2875) = 0.95765625 Calculate the square root of the right side: 0.978599126 Break this problem into two subproblems by setting (x + 0.2875) equal to 0.978599126 and -0.978599126.Subproblem 1
x + 0.2875 = 0.978599126 Simplifying x + 0.2875 = 0.978599126 Reorder the terms: 0.2875 + x = 0.978599126 Solving 0.2875 + x = 0.978599126 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2875' to each side of the equation. 0.2875 + -0.2875 + x = 0.978599126 + -0.2875 Combine like terms: 0.2875 + -0.2875 = 0.0000 0.0000 + x = 0.978599126 + -0.2875 x = 0.978599126 + -0.2875 Combine like terms: 0.978599126 + -0.2875 = 0.691099126 x = 0.691099126 Simplifying x = 0.691099126Subproblem 2
x + 0.2875 = -0.978599126 Simplifying x + 0.2875 = -0.978599126 Reorder the terms: 0.2875 + x = -0.978599126 Solving 0.2875 + x = -0.978599126 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2875' to each side of the equation. 0.2875 + -0.2875 + x = -0.978599126 + -0.2875 Combine like terms: 0.2875 + -0.2875 = 0.0000 0.0000 + x = -0.978599126 + -0.2875 x = -0.978599126 + -0.2875 Combine like terms: -0.978599126 + -0.2875 = -1.266099126 x = -1.266099126 Simplifying x = -1.266099126Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.691099126, -1.266099126}
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