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Simplifying 7a3 + 18a2 = -3a Reorder the terms: 18a2 + 7a3 = -3a Solving 18a2 + 7a3 = -3a Solving for variable 'a'. Reorder the terms: 3a + 18a2 + 7a3 = -3a + 3a Combine like terms: -3a + 3a = 0 3a + 18a2 + 7a3 = 0 Factor out the Greatest Common Factor (GCF), 'a'. a(3 + 18a + 7a2) = 0Subproblem 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 '(3 + 18a + 7a2)' equal to zero and attempt to solve: Simplifying 3 + 18a + 7a2 = 0 Solving 3 + 18a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.4285714286 + 2.571428571a + a2 = 0 Move the constant term to the right: Add '-0.4285714286' to each side of the equation. 0.4285714286 + 2.571428571a + -0.4285714286 + a2 = 0 + -0.4285714286 Reorder the terms: 0.4285714286 + -0.4285714286 + 2.571428571a + a2 = 0 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + 2.571428571a + a2 = 0 + -0.4285714286 2.571428571a + a2 = 0 + -0.4285714286 Combine like terms: 0 + -0.4285714286 = -0.4285714286 2.571428571a + a2 = -0.4285714286 The a term is 2.571428571a. Take half its coefficient (1.285714286). Square it (1.653061225) and add it to both sides. Add '1.653061225' to each side of the equation. 2.571428571a + 1.653061225 + a2 = -0.4285714286 + 1.653061225 Reorder the terms: 1.653061225 + 2.571428571a + a2 = -0.4285714286 + 1.653061225 Combine like terms: -0.4285714286 + 1.653061225 = 1.2244897964 1.653061225 + 2.571428571a + a2 = 1.2244897964 Factor a perfect square on the left side: (a + 1.285714286)(a + 1.285714286) = 1.2244897964 Calculate the square root of the right side: 1.106566671 Break this problem into two subproblems by setting (a + 1.285714286) equal to 1.106566671 and -1.106566671.Subproblem 1
a + 1.285714286 = 1.106566671 Simplifying a + 1.285714286 = 1.106566671 Reorder the terms: 1.285714286 + a = 1.106566671 Solving 1.285714286 + a = 1.106566671 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.285714286' to each side of the equation. 1.285714286 + -1.285714286 + a = 1.106566671 + -1.285714286 Combine like terms: 1.285714286 + -1.285714286 = 0.000000000 0.000000000 + a = 1.106566671 + -1.285714286 a = 1.106566671 + -1.285714286 Combine like terms: 1.106566671 + -1.285714286 = -0.179147615 a = -0.179147615 Simplifying a = -0.179147615Subproblem 2
a + 1.285714286 = -1.106566671 Simplifying a + 1.285714286 = -1.106566671 Reorder the terms: 1.285714286 + a = -1.106566671 Solving 1.285714286 + a = -1.106566671 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.285714286' to each side of the equation. 1.285714286 + -1.285714286 + a = -1.106566671 + -1.285714286 Combine like terms: 1.285714286 + -1.285714286 = 0.000000000 0.000000000 + a = -1.106566671 + -1.285714286 a = -1.106566671 + -1.285714286 Combine like terms: -1.106566671 + -1.285714286 = -2.392280957 a = -2.392280957 Simplifying a = -2.392280957Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.179147615, -2.392280957}Solution
a = {0, -0.179147615, -2.392280957}
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