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Simplifying 0.6(x + 4) * 0.5x = x + 3 Reorder the terms: 0.6(4 + x) * 0.5x = x + 3 Reorder the terms for easier multiplication: 0.6 * 0.5x(4 + x) = x + 3 Multiply 0.6 * 0.5 0.3x(4 + x) = x + 3 (4 * 0.3x + x * 0.3x) = x + 3 (1.2x + 0.3x2) = x + 3 Reorder the terms: 1.2x + 0.3x2 = 3 + x Solving 1.2x + 0.3x2 = 3 + x Solving for variable 'x'. Reorder the terms: -3 + 1.2x + -1x + 0.3x2 = 3 + x + -3 + -1x Combine like terms: 1.2x + -1x = 0.2x -3 + 0.2x + 0.3x2 = 3 + x + -3 + -1x Reorder the terms: -3 + 0.2x + 0.3x2 = 3 + -3 + x + -1x Combine like terms: 3 + -3 = 0 -3 + 0.2x + 0.3x2 = 0 + x + -1x -3 + 0.2x + 0.3x2 = x + -1x Combine like terms: x + -1x = 0 -3 + 0.2x + 0.3x2 = 0 Begin completing the square. Divide all terms by 0.3 the coefficient of the squared term: Divide each side by '0.3'. -10 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 0.6666666667x + 10 + x2 = 0 + 10 Reorder the terms: -10 + 10 + 0.6666666667x + x2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 0.6666666667x + x2 = 0 + 10 0.6666666667x + x2 = 0 + 10 Combine like terms: 0 + 10 = 10 0.6666666667x + x2 = 10 The x term is 0.6666666667x. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667x + 0.1111111112 + x2 = 10 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 10 + 0.1111111112 Combine like terms: 10 + 0.1111111112 = 10.1111111112 0.1111111112 + 0.6666666667x + x2 = 10.1111111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 10.1111111112 Calculate the square root of the right side: 3.179797338 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 3.179797338 and -3.179797338.Subproblem 1
x + 0.3333333334 = 3.179797338 Simplifying x + 0.3333333334 = 3.179797338 Reorder the terms: 0.3333333334 + x = 3.179797338 Solving 0.3333333334 + x = 3.179797338 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 3.179797338 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 3.179797338 + -0.3333333334 x = 3.179797338 + -0.3333333334 Combine like terms: 3.179797338 + -0.3333333334 = 2.8464640046 x = 2.8464640046 Simplifying x = 2.8464640046Subproblem 2
x + 0.3333333334 = -3.179797338 Simplifying x + 0.3333333334 = -3.179797338 Reorder the terms: 0.3333333334 + x = -3.179797338 Solving 0.3333333334 + x = -3.179797338 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -3.179797338 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -3.179797338 + -0.3333333334 x = -3.179797338 + -0.3333333334 Combine like terms: -3.179797338 + -0.3333333334 = -3.5131306714 x = -3.5131306714 Simplifying x = -3.5131306714Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.8464640046, -3.5131306714}
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