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Simplifying 2(3x + 1) = 5x(x + 1) Reorder the terms: 2(1 + 3x) = 5x(x + 1) (1 * 2 + 3x * 2) = 5x(x + 1) (2 + 6x) = 5x(x + 1) Reorder the terms: 2 + 6x = 5x(1 + x) 2 + 6x = (1 * 5x + x * 5x) 2 + 6x = (5x + 5x2) Solving 2 + 6x = 5x + 5x2 Solving for variable 'x'. Combine like terms: 6x + -5x = 1x 2 + 1x + -5x2 = 5x + 5x2 + -5x + -5x2 Reorder the terms: 2 + 1x + -5x2 = 5x + -5x + 5x2 + -5x2 Combine like terms: 5x + -5x = 0 2 + 1x + -5x2 = 0 + 5x2 + -5x2 2 + 1x + -5x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 2 + 1x + -5x2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -0.4 + -0.2x + x2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + -0.2x + 0.4 + x2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + -0.2x + x2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + -0.2x + x2 = 0 + 0.4 -0.2x + x2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 -0.2x + x2 = 0.4 The x term is -0.2x. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2x + 0.01 + x2 = 0.4 + 0.01 Reorder the terms: 0.01 + -0.2x + x2 = 0.4 + 0.01 Combine like terms: 0.4 + 0.01 = 0.41 0.01 + -0.2x + x2 = 0.41 Factor a perfect square on the left side: (x + -0.1)(x + -0.1) = 0.41 Calculate the square root of the right side: 0.640312424 Break this problem into two subproblems by setting (x + -0.1) equal to 0.640312424 and -0.640312424.Subproblem 1
x + -0.1 = 0.640312424 Simplifying x + -0.1 = 0.640312424 Reorder the terms: -0.1 + x = 0.640312424 Solving -0.1 + x = 0.640312424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = 0.640312424 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = 0.640312424 + 0.1 x = 0.640312424 + 0.1 Combine like terms: 0.640312424 + 0.1 = 0.740312424 x = 0.740312424 Simplifying x = 0.740312424Subproblem 2
x + -0.1 = -0.640312424 Simplifying x + -0.1 = -0.640312424 Reorder the terms: -0.1 + x = -0.640312424 Solving -0.1 + x = -0.640312424 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + x = -0.640312424 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + x = -0.640312424 + 0.1 x = -0.640312424 + 0.1 Combine like terms: -0.640312424 + 0.1 = -0.540312424 x = -0.540312424 Simplifying x = -0.540312424Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.740312424, -0.540312424}
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