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Simplifying 9a(a + 2) + 15 = 33 Reorder the terms: 9a(2 + a) + 15 = 33 (2 * 9a + a * 9a) + 15 = 33 (18a + 9a2) + 15 = 33 Reorder the terms: 15 + 18a + 9a2 = 33 Solving 15 + 18a + 9a2 = 33 Solving for variable 'a'. Reorder the terms: 15 + -33 + 18a + 9a2 = 33 + -33 Combine like terms: 15 + -33 = -18 -18 + 18a + 9a2 = 33 + -33 Combine like terms: 33 + -33 = 0 -18 + 18a + 9a2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(-2 + 2a + a2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(-2 + 2a + a2)' equal to zero and attempt to solve: Simplifying -2 + 2a + a2 = 0 Solving -2 + 2a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + 2a + 2 + a2 = 0 + 2 Reorder the terms: -2 + 2 + 2a + a2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 2a + a2 = 0 + 2 2a + a2 = 0 + 2 Combine like terms: 0 + 2 = 2 2a + a2 = 2 The a term is 2a. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2a + 1 + a2 = 2 + 1 Reorder the terms: 1 + 2a + a2 = 2 + 1 Combine like terms: 2 + 1 = 3 1 + 2a + a2 = 3 Factor a perfect square on the left side: (a + 1)(a + 1) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (a + 1) equal to 1.732050808 and -1.732050808.Subproblem 1
a + 1 = 1.732050808 Simplifying a + 1 = 1.732050808 Reorder the terms: 1 + a = 1.732050808 Solving 1 + a = 1.732050808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = 1.732050808 + -1 Combine like terms: 1 + -1 = 0 0 + a = 1.732050808 + -1 a = 1.732050808 + -1 Combine like terms: 1.732050808 + -1 = 0.732050808 a = 0.732050808 Simplifying a = 0.732050808Subproblem 2
a + 1 = -1.732050808 Simplifying a + 1 = -1.732050808 Reorder the terms: 1 + a = -1.732050808 Solving 1 + a = -1.732050808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + a = -1.732050808 + -1 Combine like terms: 1 + -1 = 0 0 + a = -1.732050808 + -1 a = -1.732050808 + -1 Combine like terms: -1.732050808 + -1 = -2.732050808 a = -2.732050808 Simplifying a = -2.732050808Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.732050808, -2.732050808}Solution
a = {0.732050808, -2.732050808}
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