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Simplifying 5x(10x + 5) + -2x(10x + 2) = 3x + 1 Reorder the terms: 5x(5 + 10x) + -2x(10x + 2) = 3x + 1 (5 * 5x + 10x * 5x) + -2x(10x + 2) = 3x + 1 (25x + 50x2) + -2x(10x + 2) = 3x + 1 Reorder the terms: 25x + 50x2 + -2x(2 + 10x) = 3x + 1 25x + 50x2 + (2 * -2x + 10x * -2x) = 3x + 1 25x + 50x2 + (-4x + -20x2) = 3x + 1 Reorder the terms: 25x + -4x + 50x2 + -20x2 = 3x + 1 Combine like terms: 25x + -4x = 21x 21x + 50x2 + -20x2 = 3x + 1 Combine like terms: 50x2 + -20x2 = 30x2 21x + 30x2 = 3x + 1 Reorder the terms: 21x + 30x2 = 1 + 3x Solving 21x + 30x2 = 1 + 3x Solving for variable 'x'. Reorder the terms: -1 + 21x + -3x + 30x2 = 1 + 3x + -1 + -3x Combine like terms: 21x + -3x = 18x -1 + 18x + 30x2 = 1 + 3x + -1 + -3x Reorder the terms: -1 + 18x + 30x2 = 1 + -1 + 3x + -3x Combine like terms: 1 + -1 = 0 -1 + 18x + 30x2 = 0 + 3x + -3x -1 + 18x + 30x2 = 3x + -3x Combine like terms: 3x + -3x = 0 -1 + 18x + 30x2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -0.03333333333 + 0.6x + x2 = 0.0 Move the constant term to the right: Add '0.03333333333' to each side of the equation. -0.03333333333 + 0.6x + 0.03333333333 + x2 = 0.0 + 0.03333333333 Reorder the terms: -0.03333333333 + 0.03333333333 + 0.6x + x2 = 0.0 + 0.03333333333 Combine like terms: -0.03333333333 + 0.03333333333 = 0.00000000000 0.00000000000 + 0.6x + x2 = 0.0 + 0.03333333333 0.6x + x2 = 0.0 + 0.03333333333 Combine like terms: 0.0 + 0.03333333333 = 0.03333333333 0.6x + x2 = 0.03333333333 The x term is 0.6x. Take half its coefficient (0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. 0.6x + 0.09 + x2 = 0.03333333333 + 0.09 Reorder the terms: 0.09 + 0.6x + x2 = 0.03333333333 + 0.09 Combine like terms: 0.03333333333 + 0.09 = 0.12333333333 0.09 + 0.6x + x2 = 0.12333333333 Factor a perfect square on the left side: (x + 0.3)(x + 0.3) = 0.12333333333 Calculate the square root of the right side: 0.351188458 Break this problem into two subproblems by setting (x + 0.3) equal to 0.351188458 and -0.351188458.Subproblem 1
x + 0.3 = 0.351188458 Simplifying x + 0.3 = 0.351188458 Reorder the terms: 0.3 + x = 0.351188458 Solving 0.3 + x = 0.351188458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + x = 0.351188458 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + x = 0.351188458 + -0.3 x = 0.351188458 + -0.3 Combine like terms: 0.351188458 + -0.3 = 0.051188458 x = 0.051188458 Simplifying x = 0.051188458Subproblem 2
x + 0.3 = -0.351188458 Simplifying x + 0.3 = -0.351188458 Reorder the terms: 0.3 + x = -0.351188458 Solving 0.3 + x = -0.351188458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + x = -0.351188458 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + x = -0.351188458 + -0.3 x = -0.351188458 + -0.3 Combine like terms: -0.351188458 + -0.3 = -0.651188458 x = -0.651188458 Simplifying x = -0.651188458Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.051188458, -0.651188458}
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