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Simplifying 3x(1 + 5x) = 0.5x + 14 (1 * 3x + 5x * 3x) = 0.5x + 14 (3x + 15x2) = 0.5x + 14 Reorder the terms: 3x + 15x2 = 14 + 0.5x Solving 3x + 15x2 = 14 + 0.5x Solving for variable 'x'. Reorder the terms: -14 + 3x + -0.5x + 15x2 = 14 + 0.5x + -14 + -0.5x Combine like terms: 3x + -0.5x = 2.5x -14 + 2.5x + 15x2 = 14 + 0.5x + -14 + -0.5x Reorder the terms: -14 + 2.5x + 15x2 = 14 + -14 + 0.5x + -0.5x Combine like terms: 14 + -14 = 0 -14 + 2.5x + 15x2 = 0 + 0.5x + -0.5x -14 + 2.5x + 15x2 = 0.5x + -0.5x Combine like terms: 0.5x + -0.5x = 0.0 -14 + 2.5x + 15x2 = 0.0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -0.9333333333 + 0.1666666667x + x2 = 0.0 Move the constant term to the right: Add '0.9333333333' to each side of the equation. -0.9333333333 + 0.1666666667x + 0.9333333333 + x2 = 0.0 + 0.9333333333 Reorder the terms: -0.9333333333 + 0.9333333333 + 0.1666666667x + x2 = 0.0 + 0.9333333333 Combine like terms: -0.9333333333 + 0.9333333333 = 0.0000000000 0.0000000000 + 0.1666666667x + x2 = 0.0 + 0.9333333333 0.1666666667x + x2 = 0.0 + 0.9333333333 Combine like terms: 0.0 + 0.9333333333 = 0.9333333333 0.1666666667x + x2 = 0.9333333333 The x term is 0.1666666667x. Take half its coefficient (0.08333333335). Square it (0.006944444447) and add it to both sides. Add '0.006944444447' to each side of the equation. 0.1666666667x + 0.006944444447 + x2 = 0.9333333333 + 0.006944444447 Reorder the terms: 0.006944444447 + 0.1666666667x + x2 = 0.9333333333 + 0.006944444447 Combine like terms: 0.9333333333 + 0.006944444447 = 0.940277777747 0.006944444447 + 0.1666666667x + x2 = 0.940277777747 Factor a perfect square on the left side: (x + 0.08333333335)(x + 0.08333333335) = 0.940277777747 Calculate the square root of the right side: 0.969679214 Break this problem into two subproblems by setting (x + 0.08333333335) equal to 0.969679214 and -0.969679214.Subproblem 1
x + 0.08333333335 = 0.969679214 Simplifying x + 0.08333333335 = 0.969679214 Reorder the terms: 0.08333333335 + x = 0.969679214 Solving 0.08333333335 + x = 0.969679214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = 0.969679214 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = 0.969679214 + -0.08333333335 x = 0.969679214 + -0.08333333335 Combine like terms: 0.969679214 + -0.08333333335 = 0.88634588065 x = 0.88634588065 Simplifying x = 0.88634588065Subproblem 2
x + 0.08333333335 = -0.969679214 Simplifying x + 0.08333333335 = -0.969679214 Reorder the terms: 0.08333333335 + x = -0.969679214 Solving 0.08333333335 + x = -0.969679214 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + x = -0.969679214 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + x = -0.969679214 + -0.08333333335 x = -0.969679214 + -0.08333333335 Combine like terms: -0.969679214 + -0.08333333335 = -1.05301254735 x = -1.05301254735 Simplifying x = -1.05301254735Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.88634588065, -1.05301254735}
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