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Simplifying 3x(2x + 1) = 1 Reorder the terms: 3x(1 + 2x) = 1 (1 * 3x + 2x * 3x) = 1 (3x + 6x2) = 1 Solving 3x + 6x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 3x + 6x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 3x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.1666666667 + 0.5x + x2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.5x + 0.1666666667 + x2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + 0.5x + x2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + 0.5x + x2 = 0 + 0.1666666667 0.5x + x2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 0.5x + x2 = 0.1666666667 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 0.1666666667 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 0.1666666667 + 0.0625 Combine like terms: 0.1666666667 + 0.0625 = 0.2291666667 0.0625 + 0.5x + x2 = 0.2291666667 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 0.2291666667 Calculate the square root of the right side: 0.478713554 Break this problem into two subproblems by setting (x + 0.25) equal to 0.478713554 and -0.478713554.Subproblem 1
x + 0.25 = 0.478713554 Simplifying x + 0.25 = 0.478713554 Reorder the terms: 0.25 + x = 0.478713554 Solving 0.25 + x = 0.478713554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.478713554 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.478713554 + -0.25 x = 0.478713554 + -0.25 Combine like terms: 0.478713554 + -0.25 = 0.228713554 x = 0.228713554 Simplifying x = 0.228713554Subproblem 2
x + 0.25 = -0.478713554 Simplifying x + 0.25 = -0.478713554 Reorder the terms: 0.25 + x = -0.478713554 Solving 0.25 + x = -0.478713554 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.478713554 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.478713554 + -0.25 x = -0.478713554 + -0.25 Combine like terms: -0.478713554 + -0.25 = -0.728713554 x = -0.728713554 Simplifying x = -0.728713554Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.228713554, -0.728713554}
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