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Simplifying 9x2 + -6x + -32 = 0 Reorder the terms: -32 + -6x + 9x2 = 0 Solving -32 + -6x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -3.555555556 + -0.6666666667x + x2 = 0 Move the constant term to the right: Add '3.555555556' to each side of the equation. -3.555555556 + -0.6666666667x + 3.555555556 + x2 = 0 + 3.555555556 Reorder the terms: -3.555555556 + 3.555555556 + -0.6666666667x + x2 = 0 + 3.555555556 Combine like terms: -3.555555556 + 3.555555556 = 0.000000000 0.000000000 + -0.6666666667x + x2 = 0 + 3.555555556 -0.6666666667x + x2 = 0 + 3.555555556 Combine like terms: 0 + 3.555555556 = 3.555555556 -0.6666666667x + x2 = 3.555555556 The x term is -0.6666666667x. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667x + 0.1111111112 + x2 = 3.555555556 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667x + x2 = 3.555555556 + 0.1111111112 Combine like terms: 3.555555556 + 0.1111111112 = 3.6666666672 0.1111111112 + -0.6666666667x + x2 = 3.6666666672 Factor a perfect square on the left side: (x + -0.3333333334)(x + -0.3333333334) = 3.6666666672 Calculate the square root of the right side: 1.914854216 Break this problem into two subproblems by setting (x + -0.3333333334) equal to 1.914854216 and -1.914854216.Subproblem 1
x + -0.3333333334 = 1.914854216 Simplifying x + -0.3333333334 = 1.914854216 Reorder the terms: -0.3333333334 + x = 1.914854216 Solving -0.3333333334 + x = 1.914854216 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 1.914854216 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 1.914854216 + 0.3333333334 x = 1.914854216 + 0.3333333334 Combine like terms: 1.914854216 + 0.3333333334 = 2.2481875494 x = 2.2481875494 Simplifying x = 2.2481875494Subproblem 2
x + -0.3333333334 = -1.914854216 Simplifying x + -0.3333333334 = -1.914854216 Reorder the terms: -0.3333333334 + x = -1.914854216 Solving -0.3333333334 + x = -1.914854216 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -1.914854216 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -1.914854216 + 0.3333333334 x = -1.914854216 + 0.3333333334 Combine like terms: -1.914854216 + 0.3333333334 = -1.5815208826 x = -1.5815208826 Simplifying x = -1.5815208826Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.2481875494, -1.5815208826}
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