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Simplifying 81x2 + -54x + -9 = 0 Reorder the terms: -9 + -54x + 81x2 = 0 Solving -9 + -54x + 81x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '9'. 9(-1 + -6x + 9x2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(-1 + -6x + 9x2)' equal to zero and attempt to solve: Simplifying -1 + -6x + 9x2 = 0 Solving -1 + -6x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -0.6666666667x + x2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -0.6666666667x + 0.1111111111 + x2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -0.6666666667x + x2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -0.6666666667x + x2 = 0 + 0.1111111111 -0.6666666667x + x2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -0.6666666667x + x2 = 0.1111111111 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 = 0.1111111111 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667x + x2 = 0.1111111111 + 0.1111111112 Combine like terms: 0.1111111111 + 0.1111111112 = 0.2222222223 0.1111111112 + -0.6666666667x + x2 = 0.2222222223 Factor a perfect square on the left side: (x + -0.3333333334)(x + -0.3333333334) = 0.2222222223 Calculate the square root of the right side: 0.471404521 Break this problem into two subproblems by setting (x + -0.3333333334) equal to 0.471404521 and -0.471404521.Subproblem 1
x + -0.3333333334 = 0.471404521 Simplifying x + -0.3333333334 = 0.471404521 Reorder the terms: -0.3333333334 + x = 0.471404521 Solving -0.3333333334 + x = 0.471404521 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 = 0.471404521 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 0.471404521 + 0.3333333334 x = 0.471404521 + 0.3333333334 Combine like terms: 0.471404521 + 0.3333333334 = 0.8047378544 x = 0.8047378544 Simplifying x = 0.8047378544Subproblem 2
x + -0.3333333334 = -0.471404521 Simplifying x + -0.3333333334 = -0.471404521 Reorder the terms: -0.3333333334 + x = -0.471404521 Solving -0.3333333334 + x = -0.471404521 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 = -0.471404521 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -0.471404521 + 0.3333333334 x = -0.471404521 + 0.3333333334 Combine like terms: -0.471404521 + 0.3333333334 = -0.1380711876 x = -0.1380711876 Simplifying x = -0.1380711876Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8047378544, -0.1380711876}Solution
x = {0.8047378544, -0.1380711876}
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