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Simplifying 6x(2x + 3) = 1 Reorder the terms: 6x(3 + 2x) = 1 (3 * 6x + 2x * 6x) = 1 (18x + 12x2) = 1 Solving 18x + 12x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 18x + 12x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 18x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.08333333333 + 1.5x + x2 = 0 Move the constant term to the right: Add '0.08333333333' to each side of the equation. -0.08333333333 + 1.5x + 0.08333333333 + x2 = 0 + 0.08333333333 Reorder the terms: -0.08333333333 + 0.08333333333 + 1.5x + x2 = 0 + 0.08333333333 Combine like terms: -0.08333333333 + 0.08333333333 = 0.00000000000 0.00000000000 + 1.5x + x2 = 0 + 0.08333333333 1.5x + x2 = 0 + 0.08333333333 Combine like terms: 0 + 0.08333333333 = 0.08333333333 1.5x + x2 = 0.08333333333 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 0.08333333333 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 0.08333333333 + 0.5625 Combine like terms: 0.08333333333 + 0.5625 = 0.64583333333 0.5625 + 1.5x + x2 = 0.64583333333 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 0.64583333333 Calculate the square root of the right side: 0.803637563 Break this problem into two subproblems by setting (x + 0.75) equal to 0.803637563 and -0.803637563.Subproblem 1
x + 0.75 = 0.803637563 Simplifying x + 0.75 = 0.803637563 Reorder the terms: 0.75 + x = 0.803637563 Solving 0.75 + x = 0.803637563 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 0.803637563 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 0.803637563 + -0.75 x = 0.803637563 + -0.75 Combine like terms: 0.803637563 + -0.75 = 0.053637563 x = 0.053637563 Simplifying x = 0.053637563Subproblem 2
x + 0.75 = -0.803637563 Simplifying x + 0.75 = -0.803637563 Reorder the terms: 0.75 + x = -0.803637563 Solving 0.75 + x = -0.803637563 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -0.803637563 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -0.803637563 + -0.75 x = -0.803637563 + -0.75 Combine like terms: -0.803637563 + -0.75 = -1.553637563 x = -1.553637563 Simplifying x = -1.553637563Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.053637563, -1.553637563}
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