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Simplifying 27 = 3x(4x + 1)(0.5) Reorder the terms: 27 = 3x(1 + 4x)(0.5) Reorder the terms for easier multiplication: 27 = 3 * 0.5x(1 + 4x) Multiply 3 * 0.5 27 = 1.5x(1 + 4x) 27 = (1 * 1.5x + 4x * 1.5x) 27 = (1.5x + 6x2) Solving 27 = 1.5x + 6x2 Solving for variable 'x'. Reorder the terms: 27 + -1.5x + -6x2 = 1.5x + -1.5x + 6x2 + -6x2 Combine like terms: 1.5x + -1.5x = 0.0 27 + -1.5x + -6x2 = 0.0 + 6x2 + -6x2 27 + -1.5x + -6x2 = 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 0 27 + -1.5x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -4.5 + 0.25x + x2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 0.25x + 4.5 + x2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 0.25x + x2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 0.25x + x2 = 0 + 4.5 0.25x + x2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 0.25x + x2 = 4.5 The x term is 0.25x. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25x + 0.015625 + x2 = 4.5 + 0.015625 Reorder the terms: 0.015625 + 0.25x + x2 = 4.5 + 0.015625 Combine like terms: 4.5 + 0.015625 = 4.515625 0.015625 + 0.25x + x2 = 4.515625 Factor a perfect square on the left side: (x + 0.125)(x + 0.125) = 4.515625 Calculate the square root of the right side: 2.125 Break this problem into two subproblems by setting (x + 0.125) equal to 2.125 and -2.125.Subproblem 1
x + 0.125 = 2.125 Simplifying x + 0.125 = 2.125 Reorder the terms: 0.125 + x = 2.125 Solving 0.125 + x = 2.125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = 2.125 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = 2.125 + -0.125 x = 2.125 + -0.125 Combine like terms: 2.125 + -0.125 = 2 x = 2 Simplifying x = 2Subproblem 2
x + 0.125 = -2.125 Simplifying x + 0.125 = -2.125 Reorder the terms: 0.125 + x = -2.125 Solving 0.125 + x = -2.125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = -2.125 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = -2.125 + -0.125 x = -2.125 + -0.125 Combine like terms: -2.125 + -0.125 = -2.25 x = -2.25 Simplifying x = -2.25Solution
The solution to the problem is based on the solutions from the subproblems. x = {2, -2.25}
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