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Simplifying 6x(3x + 2) + -5(2x + 3) = 0 Reorder the terms: 6x(2 + 3x) + -5(2x + 3) = 0 (2 * 6x + 3x * 6x) + -5(2x + 3) = 0 (12x + 18x2) + -5(2x + 3) = 0 Reorder the terms: 12x + 18x2 + -5(3 + 2x) = 0 12x + 18x2 + (3 * -5 + 2x * -5) = 0 12x + 18x2 + (-15 + -10x) = 0 Reorder the terms: -15 + 12x + -10x + 18x2 = 0 Combine like terms: 12x + -10x = 2x -15 + 2x + 18x2 = 0 Solving -15 + 2x + 18x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -0.8333333333 + 0.1111111111x + x2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + 0.1111111111x + 0.8333333333 + x2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + 0.1111111111x + x2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + 0.1111111111x + x2 = 0 + 0.8333333333 0.1111111111x + x2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 0.1111111111x + x2 = 0.8333333333 The x term is 0.1111111111x. Take half its coefficient (0.05555555555). Square it (0.003086419752) and add it to both sides. Add '0.003086419752' to each side of the equation. 0.1111111111x + 0.003086419752 + x2 = 0.8333333333 + 0.003086419752 Reorder the terms: 0.003086419752 + 0.1111111111x + x2 = 0.8333333333 + 0.003086419752 Combine like terms: 0.8333333333 + 0.003086419752 = 0.836419753052 0.003086419752 + 0.1111111111x + x2 = 0.836419753052 Factor a perfect square on the left side: (x + 0.05555555555)(x + 0.05555555555) = 0.836419753052 Calculate the square root of the right side: 0.914559868 Break this problem into two subproblems by setting (x + 0.05555555555) equal to 0.914559868 and -0.914559868.Subproblem 1
x + 0.05555555555 = 0.914559868 Simplifying x + 0.05555555555 = 0.914559868 Reorder the terms: 0.05555555555 + x = 0.914559868 Solving 0.05555555555 + x = 0.914559868 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.05555555555' to each side of the equation. 0.05555555555 + -0.05555555555 + x = 0.914559868 + -0.05555555555 Combine like terms: 0.05555555555 + -0.05555555555 = 0.00000000000 0.00000000000 + x = 0.914559868 + -0.05555555555 x = 0.914559868 + -0.05555555555 Combine like terms: 0.914559868 + -0.05555555555 = 0.85900431245 x = 0.85900431245 Simplifying x = 0.85900431245Subproblem 2
x + 0.05555555555 = -0.914559868 Simplifying x + 0.05555555555 = -0.914559868 Reorder the terms: 0.05555555555 + x = -0.914559868 Solving 0.05555555555 + x = -0.914559868 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.05555555555' to each side of the equation. 0.05555555555 + -0.05555555555 + x = -0.914559868 + -0.05555555555 Combine like terms: 0.05555555555 + -0.05555555555 = 0.00000000000 0.00000000000 + x = -0.914559868 + -0.05555555555 x = -0.914559868 + -0.05555555555 Combine like terms: -0.914559868 + -0.05555555555 = -0.97011542355 x = -0.97011542355 Simplifying x = -0.97011542355Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.85900431245, -0.97011542355}
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