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Simplifying 0.4(3.2x + 2) * x = 2x + 1.8 Reorder the terms: 0.4(2 + 3.2x) * x = 2x + 1.8 Reorder the terms for easier multiplication: 0.4x(2 + 3.2x) = 2x + 1.8 (2 * 0.4x + 3.2x * 0.4x) = 2x + 1.8 (0.8x + 1.28x2) = 2x + 1.8 Reorder the terms: 0.8x + 1.28x2 = 1.8 + 2x Solving 0.8x + 1.28x2 = 1.8 + 2x Solving for variable 'x'. Reorder the terms: -1.8 + 0.8x + -2x + 1.28x2 = 1.8 + 2x + -1.8 + -2x Combine like terms: 0.8x + -2x = -1.2x -1.8 + -1.2x + 1.28x2 = 1.8 + 2x + -1.8 + -2x Reorder the terms: -1.8 + -1.2x + 1.28x2 = 1.8 + -1.8 + 2x + -2x Combine like terms: 1.8 + -1.8 = 0.0 -1.8 + -1.2x + 1.28x2 = 0.0 + 2x + -2x -1.8 + -1.2x + 1.28x2 = 2x + -2x Combine like terms: 2x + -2x = 0 -1.8 + -1.2x + 1.28x2 = 0 Begin completing the square. Divide all terms by 1.28 the coefficient of the squared term: Divide each side by '1.28'. -1.40625 + -0.9375x + x2 = 0 Move the constant term to the right: Add '1.40625' to each side of the equation. -1.40625 + -0.9375x + 1.40625 + x2 = 0 + 1.40625 Reorder the terms: -1.40625 + 1.40625 + -0.9375x + x2 = 0 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + -0.9375x + x2 = 0 + 1.40625 -0.9375x + x2 = 0 + 1.40625 Combine like terms: 0 + 1.40625 = 1.40625 -0.9375x + x2 = 1.40625 The x term is -0.9375x. Take half its coefficient (-0.46875). Square it (0.2197265625) and add it to both sides. Add '0.2197265625' to each side of the equation. -0.9375x + 0.2197265625 + x2 = 1.40625 + 0.2197265625 Reorder the terms: 0.2197265625 + -0.9375x + x2 = 1.40625 + 0.2197265625 Combine like terms: 1.40625 + 0.2197265625 = 1.6259765625 0.2197265625 + -0.9375x + x2 = 1.6259765625 Factor a perfect square on the left side: (x + -0.46875)(x + -0.46875) = 1.6259765625 Calculate the square root of the right side: 1.27513786 Break this problem into two subproblems by setting (x + -0.46875) equal to 1.27513786 and -1.27513786.Subproblem 1
x + -0.46875 = 1.27513786 Simplifying x + -0.46875 = 1.27513786 Reorder the terms: -0.46875 + x = 1.27513786 Solving -0.46875 + x = 1.27513786 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.46875' to each side of the equation. -0.46875 + 0.46875 + x = 1.27513786 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + x = 1.27513786 + 0.46875 x = 1.27513786 + 0.46875 Combine like terms: 1.27513786 + 0.46875 = 1.74388786 x = 1.74388786 Simplifying x = 1.74388786Subproblem 2
x + -0.46875 = -1.27513786 Simplifying x + -0.46875 = -1.27513786 Reorder the terms: -0.46875 + x = -1.27513786 Solving -0.46875 + x = -1.27513786 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.46875' to each side of the equation. -0.46875 + 0.46875 + x = -1.27513786 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + x = -1.27513786 + 0.46875 x = -1.27513786 + 0.46875 Combine like terms: -1.27513786 + 0.46875 = -0.80638786 x = -0.80638786 Simplifying x = -0.80638786Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.74388786, -0.80638786}
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