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Simplifying 27 = -9a(a + 5) Reorder the terms: 27 = -9a(5 + a) 27 = (5 * -9a + a * -9a) 27 = (-45a + -9a2) Solving 27 = -45a + -9a2 Solving for variable 'a'. Reorder the terms: 27 + 45a + 9a2 = -45a + 45a + -9a2 + 9a2 Combine like terms: -45a + 45a = 0 27 + 45a + 9a2 = 0 + -9a2 + 9a2 27 + 45a + 9a2 = -9a2 + 9a2 Combine like terms: -9a2 + 9a2 = 0 27 + 45a + 9a2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(3 + 5a + a2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(3 + 5a + a2)' equal to zero and attempt to solve: Simplifying 3 + 5a + a2 = 0 Solving 3 + 5a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + 5a + -3 + a2 = 0 + -3 Reorder the terms: 3 + -3 + 5a + a2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 5a + a2 = 0 + -3 5a + a2 = 0 + -3 Combine like terms: 0 + -3 = -3 5a + a2 = -3 The a term is 5a. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5a + 6.25 + a2 = -3 + 6.25 Reorder the terms: 6.25 + 5a + a2 = -3 + 6.25 Combine like terms: -3 + 6.25 = 3.25 6.25 + 5a + a2 = 3.25 Factor a perfect square on the left side: (a + 2.5)(a + 2.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (a + 2.5) equal to 1.802775638 and -1.802775638.Subproblem 1
a + 2.5 = 1.802775638 Simplifying a + 2.5 = 1.802775638 Reorder the terms: 2.5 + a = 1.802775638 Solving 2.5 + a = 1.802775638 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + a = 1.802775638 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = 1.802775638 + -2.5 a = 1.802775638 + -2.5 Combine like terms: 1.802775638 + -2.5 = -0.697224362 a = -0.697224362 Simplifying a = -0.697224362Subproblem 2
a + 2.5 = -1.802775638 Simplifying a + 2.5 = -1.802775638 Reorder the terms: 2.5 + a = -1.802775638 Solving 2.5 + a = -1.802775638 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + a = -1.802775638 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + a = -1.802775638 + -2.5 a = -1.802775638 + -2.5 Combine like terms: -1.802775638 + -2.5 = -4.302775638 a = -4.302775638 Simplifying a = -4.302775638Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.697224362, -4.302775638}Solution
a = {-0.697224362, -4.302775638}
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