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Simplifying 3x + -1[5x(9x + 3) + -11] = 0 Reorder the terms: 3x + -1[5x(3 + 9x) + -11] = 0 3x + -1[(3 * 5x + 9x * 5x) + -11] = 0 3x + -1[(15x + 45x2) + -11] = 0 Reorder the terms: 3x + -1[-11 + 15x + 45x2] = 0 3x + [-11 * -1 + 15x * -1 + 45x2 * -1] = 0 3x + [11 + -15x + -45x2] = 0 Reorder the terms: 11 + 3x + -15x + -45x2 = 0 Combine like terms: 3x + -15x = -12x 11 + -12x + -45x2 = 0 Solving 11 + -12x + -45x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -45 the coefficient of the squared term: Divide each side by '-45'. -0.2444444444 + 0.2666666667x + x2 = 0 Move the constant term to the right: Add '0.2444444444' to each side of the equation. -0.2444444444 + 0.2666666667x + 0.2444444444 + x2 = 0 + 0.2444444444 Reorder the terms: -0.2444444444 + 0.2444444444 + 0.2666666667x + x2 = 0 + 0.2444444444 Combine like terms: -0.2444444444 + 0.2444444444 = 0.0000000000 0.0000000000 + 0.2666666667x + x2 = 0 + 0.2444444444 0.2666666667x + x2 = 0 + 0.2444444444 Combine like terms: 0 + 0.2444444444 = 0.2444444444 0.2666666667x + x2 = 0.2444444444 The x term is 0.2666666667x. Take half its coefficient (0.1333333334). Square it (0.01777777780) and add it to both sides. Add '0.01777777780' to each side of the equation. 0.2666666667x + 0.01777777780 + x2 = 0.2444444444 + 0.01777777780 Reorder the terms: 0.01777777780 + 0.2666666667x + x2 = 0.2444444444 + 0.01777777780 Combine like terms: 0.2444444444 + 0.01777777780 = 0.2622222222 0.01777777780 + 0.2666666667x + x2 = 0.2622222222 Factor a perfect square on the left side: (x + 0.1333333334)(x + 0.1333333334) = 0.2622222222 Calculate the square root of the right side: 0.512076383 Break this problem into two subproblems by setting (x + 0.1333333334) equal to 0.512076383 and -0.512076383.Subproblem 1
x + 0.1333333334 = 0.512076383 Simplifying x + 0.1333333334 = 0.512076383 Reorder the terms: 0.1333333334 + x = 0.512076383 Solving 0.1333333334 + x = 0.512076383 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1333333334' to each side of the equation. 0.1333333334 + -0.1333333334 + x = 0.512076383 + -0.1333333334 Combine like terms: 0.1333333334 + -0.1333333334 = 0.0000000000 0.0000000000 + x = 0.512076383 + -0.1333333334 x = 0.512076383 + -0.1333333334 Combine like terms: 0.512076383 + -0.1333333334 = 0.3787430496 x = 0.3787430496 Simplifying x = 0.3787430496Subproblem 2
x + 0.1333333334 = -0.512076383 Simplifying x + 0.1333333334 = -0.512076383 Reorder the terms: 0.1333333334 + x = -0.512076383 Solving 0.1333333334 + x = -0.512076383 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1333333334' to each side of the equation. 0.1333333334 + -0.1333333334 + x = -0.512076383 + -0.1333333334 Combine like terms: 0.1333333334 + -0.1333333334 = 0.0000000000 0.0000000000 + x = -0.512076383 + -0.1333333334 x = -0.512076383 + -0.1333333334 Combine like terms: -0.512076383 + -0.1333333334 = -0.6454097164 x = -0.6454097164 Simplifying x = -0.6454097164Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.3787430496, -0.6454097164}
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