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Simplifying 2a(a + 3) = 210 Reorder the terms: 2a(3 + a) = 210 (3 * 2a + a * 2a) = 210 (6a + 2a2) = 210 Solving 6a + 2a2 = 210 Solving for variable 'a'. Reorder the terms: -210 + 6a + 2a2 = 210 + -210 Combine like terms: 210 + -210 = 0 -210 + 6a + 2a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-105 + 3a + a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-105 + 3a + a2)' equal to zero and attempt to solve: Simplifying -105 + 3a + a2 = 0 Solving -105 + 3a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '105' to each side of the equation. -105 + 3a + 105 + a2 = 0 + 105 Reorder the terms: -105 + 105 + 3a + a2 = 0 + 105 Combine like terms: -105 + 105 = 0 0 + 3a + a2 = 0 + 105 3a + a2 = 0 + 105 Combine like terms: 0 + 105 = 105 3a + a2 = 105 The a term is 3a. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3a + 2.25 + a2 = 105 + 2.25 Reorder the terms: 2.25 + 3a + a2 = 105 + 2.25 Combine like terms: 105 + 2.25 = 107.25 2.25 + 3a + a2 = 107.25 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 107.25 Calculate the square root of the right side: 10.356157589 Break this problem into two subproblems by setting (a + 1.5) equal to 10.356157589 and -10.356157589.Subproblem 1
a + 1.5 = 10.356157589 Simplifying a + 1.5 = 10.356157589 Reorder the terms: 1.5 + a = 10.356157589 Solving 1.5 + a = 10.356157589 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 10.356157589 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 10.356157589 + -1.5 a = 10.356157589 + -1.5 Combine like terms: 10.356157589 + -1.5 = 8.856157589 a = 8.856157589 Simplifying a = 8.856157589Subproblem 2
a + 1.5 = -10.356157589 Simplifying a + 1.5 = -10.356157589 Reorder the terms: 1.5 + a = -10.356157589 Solving 1.5 + a = -10.356157589 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -10.356157589 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -10.356157589 + -1.5 a = -10.356157589 + -1.5 Combine like terms: -10.356157589 + -1.5 = -11.856157589 a = -11.856157589 Simplifying a = -11.856157589Solution
The solution to the problem is based on the solutions from the subproblems. a = {8.856157589, -11.856157589}Solution
a = {8.856157589, -11.856157589}
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