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Simplifying 0 = 36x2 + -24x + 1 Reorder the terms: 0 = 1 + -24x + 36x2 Solving 0 = 1 + -24x + 36x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + 24x + -36x2 = 1 + -24x + 36x2 + -1 + 24x + -36x2 Reorder the terms: -1 + 24x + -36x2 = 1 + -1 + -24x + 24x + 36x2 + -36x2 Combine like terms: 1 + -1 = 0 -1 + 24x + -36x2 = 0 + -24x + 24x + 36x2 + -36x2 -1 + 24x + -36x2 = -24x + 24x + 36x2 + -36x2 Combine like terms: -24x + 24x = 0 -1 + 24x + -36x2 = 0 + 36x2 + -36x2 -1 + 24x + -36x2 = 36x2 + -36x2 Combine like terms: 36x2 + -36x2 = 0 -1 + 24x + -36x2 = 0 Begin completing the square. Divide all terms by -36 the coefficient of the squared term: Divide each side by '-36'. 0.02777777778 + -0.6666666667x + x2 = 0 Move the constant term to the right: Add '-0.02777777778' to each side of the equation. 0.02777777778 + -0.6666666667x + -0.02777777778 + x2 = 0 + -0.02777777778 Reorder the terms: 0.02777777778 + -0.02777777778 + -0.6666666667x + x2 = 0 + -0.02777777778 Combine like terms: 0.02777777778 + -0.02777777778 = 0.00000000000 0.00000000000 + -0.6666666667x + x2 = 0 + -0.02777777778 -0.6666666667x + x2 = 0 + -0.02777777778 Combine like terms: 0 + -0.02777777778 = -0.02777777778 -0.6666666667x + x2 = -0.02777777778 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 = -0.02777777778 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667x + x2 = -0.02777777778 + 0.1111111112 Combine like terms: -0.02777777778 + 0.1111111112 = 0.08333333342 0.1111111112 + -0.6666666667x + x2 = 0.08333333342 Factor a perfect square on the left side: (x + -0.3333333334)(x + -0.3333333334) = 0.08333333342 Calculate the square root of the right side: 0.288675135 Break this problem into two subproblems by setting (x + -0.3333333334) equal to 0.288675135 and -0.288675135.Subproblem 1
x + -0.3333333334 = 0.288675135 Simplifying x + -0.3333333334 = 0.288675135 Reorder the terms: -0.3333333334 + x = 0.288675135 Solving -0.3333333334 + x = 0.288675135 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 = 0.288675135 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 0.288675135 + 0.3333333334 x = 0.288675135 + 0.3333333334 Combine like terms: 0.288675135 + 0.3333333334 = 0.6220084684 x = 0.6220084684 Simplifying x = 0.6220084684Subproblem 2
x + -0.3333333334 = -0.288675135 Simplifying x + -0.3333333334 = -0.288675135 Reorder the terms: -0.3333333334 + x = -0.288675135 Solving -0.3333333334 + x = -0.288675135 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 = -0.288675135 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -0.288675135 + 0.3333333334 x = -0.288675135 + 0.3333333334 Combine like terms: -0.288675135 + 0.3333333334 = 0.0446581984 x = 0.0446581984 Simplifying x = 0.0446581984Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6220084684, 0.0446581984}
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