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Simplifying 6x(3x + -5) + 12 = x Reorder the terms: 6x(-5 + 3x) + 12 = x (-5 * 6x + 3x * 6x) + 12 = x (-30x + 18x2) + 12 = x Reorder the terms: 12 + -30x + 18x2 = x Solving 12 + -30x + 18x2 = x Solving for variable 'x'. Reorder the terms: 12 + -30x + -1x + 18x2 = x + -1x Combine like terms: -30x + -1x = -31x 12 + -31x + 18x2 = x + -1x Combine like terms: x + -1x = 0 12 + -31x + 18x2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.6666666667 + -1.722222222x + x2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -1.722222222x + -0.6666666667 + x2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -1.722222222x + x2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -1.722222222x + x2 = 0 + -0.6666666667 -1.722222222x + x2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -1.722222222x + x2 = -0.6666666667 The x term is -1.722222222x. Take half its coefficient (-0.861111111). Square it (0.7415123455) and add it to both sides. Add '0.7415123455' to each side of the equation. -1.722222222x + 0.7415123455 + x2 = -0.6666666667 + 0.7415123455 Reorder the terms: 0.7415123455 + -1.722222222x + x2 = -0.6666666667 + 0.7415123455 Combine like terms: -0.6666666667 + 0.7415123455 = 0.0748456788 0.7415123455 + -1.722222222x + x2 = 0.0748456788 Factor a perfect square on the left side: (x + -0.861111111)(x + -0.861111111) = 0.0748456788 Calculate the square root of the right side: 0.273579383 Break this problem into two subproblems by setting (x + -0.861111111) equal to 0.273579383 and -0.273579383.Subproblem 1
x + -0.861111111 = 0.273579383 Simplifying x + -0.861111111 = 0.273579383 Reorder the terms: -0.861111111 + x = 0.273579383 Solving -0.861111111 + x = 0.273579383 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.861111111' to each side of the equation. -0.861111111 + 0.861111111 + x = 0.273579383 + 0.861111111 Combine like terms: -0.861111111 + 0.861111111 = 0.000000000 0.000000000 + x = 0.273579383 + 0.861111111 x = 0.273579383 + 0.861111111 Combine like terms: 0.273579383 + 0.861111111 = 1.134690494 x = 1.134690494 Simplifying x = 1.134690494Subproblem 2
x + -0.861111111 = -0.273579383 Simplifying x + -0.861111111 = -0.273579383 Reorder the terms: -0.861111111 + x = -0.273579383 Solving -0.861111111 + x = -0.273579383 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.861111111' to each side of the equation. -0.861111111 + 0.861111111 + x = -0.273579383 + 0.861111111 Combine like terms: -0.861111111 + 0.861111111 = 0.000000000 0.000000000 + x = -0.273579383 + 0.861111111 x = -0.273579383 + 0.861111111 Combine like terms: -0.273579383 + 0.861111111 = 0.587531728 x = 0.587531728 Simplifying x = 0.587531728Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.134690494, 0.587531728}
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