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Simplifying 27a3 + 18a2 = 3a Reorder the terms: 18a2 + 27a3 = 3a Solving 18a2 + 27a3 = 3a Solving for variable 'a'. Reorder the terms: -3a + 18a2 + 27a3 = 3a + -3a Combine like terms: 3a + -3a = 0 -3a + 18a2 + 27a3 = 0 Factor out the Greatest Common Factor (GCF), '3a'. 3a(-1 + 6a + 9a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor 'a' equal to zero and attempt to solve: Simplifying a = 0 Solving a = 0 Move all terms containing a to the left, all other terms to the right. Simplifying a = 0Subproblem 2
Set the factor '(-1 + 6a + 9a2)' equal to zero and attempt to solve: Simplifying -1 + 6a + 9a2 = 0 Solving -1 + 6a + 9a2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + 0.6666666667a + a2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.6666666667a + 0.1111111111 + a2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + 0.6666666667a + a2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + 0.6666666667a + a2 = 0 + 0.1111111111 0.6666666667a + a2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 0.6666666667a + a2 = 0.1111111111 The a term is 0.6666666667a. 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. 0.6666666667a + 0.1111111112 + a2 = 0.1111111111 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667a + a2 = 0.1111111111 + 0.1111111112 Combine like terms: 0.1111111111 + 0.1111111112 = 0.2222222223 0.1111111112 + 0.6666666667a + a2 = 0.2222222223 Factor a perfect square on the left side: (a + 0.3333333334)(a + 0.3333333334) = 0.2222222223 Calculate the square root of the right side: 0.471404521 Break this problem into two subproblems by setting (a + 0.3333333334) equal to 0.471404521 and -0.471404521.Subproblem 1
a + 0.3333333334 = 0.471404521 Simplifying a + 0.3333333334 = 0.471404521 Reorder the terms: 0.3333333334 + a = 0.471404521 Solving 0.3333333334 + a = 0.471404521 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + a = 0.471404521 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + a = 0.471404521 + -0.3333333334 a = 0.471404521 + -0.3333333334 Combine like terms: 0.471404521 + -0.3333333334 = 0.1380711876 a = 0.1380711876 Simplifying a = 0.1380711876Subproblem 2
a + 0.3333333334 = -0.471404521 Simplifying a + 0.3333333334 = -0.471404521 Reorder the terms: 0.3333333334 + a = -0.471404521 Solving 0.3333333334 + a = -0.471404521 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + a = -0.471404521 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + a = -0.471404521 + -0.3333333334 a = -0.471404521 + -0.3333333334 Combine like terms: -0.471404521 + -0.3333333334 = -0.8047378544 a = -0.8047378544 Simplifying a = -0.8047378544Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.1380711876, -0.8047378544}Solution
a = {0, 0.1380711876, -0.8047378544}
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