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Simplifying 2 + a(a + 30) = 180 Reorder the terms: 2 + a(30 + a) = 180 2 + (30 * a + a * a) = 180 2 + (30a + a2) = 180 Solving 2 + 30a + a2 = 180 Solving for variable 'a'. Reorder the terms: 2 + -180 + 30a + a2 = 180 + -180 Combine like terms: 2 + -180 = -178 -178 + 30a + a2 = 180 + -180 Combine like terms: 180 + -180 = 0 -178 + 30a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '178' to each side of the equation. -178 + 30a + 178 + a2 = 0 + 178 Reorder the terms: -178 + 178 + 30a + a2 = 0 + 178 Combine like terms: -178 + 178 = 0 0 + 30a + a2 = 0 + 178 30a + a2 = 0 + 178 Combine like terms: 0 + 178 = 178 30a + a2 = 178 The a term is 30a. Take half its coefficient (15). Square it (225) and add it to both sides. Add '225' to each side of the equation. 30a + 225 + a2 = 178 + 225 Reorder the terms: 225 + 30a + a2 = 178 + 225 Combine like terms: 178 + 225 = 403 225 + 30a + a2 = 403 Factor a perfect square on the left side: (a + 15)(a + 15) = 403 Calculate the square root of the right side: 20.0748599 Break this problem into two subproblems by setting (a + 15) equal to 20.0748599 and -20.0748599.Subproblem 1
a + 15 = 20.0748599 Simplifying a + 15 = 20.0748599 Reorder the terms: 15 + a = 20.0748599 Solving 15 + a = 20.0748599 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + a = 20.0748599 + -15 Combine like terms: 15 + -15 = 0 0 + a = 20.0748599 + -15 a = 20.0748599 + -15 Combine like terms: 20.0748599 + -15 = 5.0748599 a = 5.0748599 Simplifying a = 5.0748599Subproblem 2
a + 15 = -20.0748599 Simplifying a + 15 = -20.0748599 Reorder the terms: 15 + a = -20.0748599 Solving 15 + a = -20.0748599 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + a = -20.0748599 + -15 Combine like terms: 15 + -15 = 0 0 + a = -20.0748599 + -15 a = -20.0748599 + -15 Combine like terms: -20.0748599 + -15 = -35.0748599 a = -35.0748599 Simplifying a = -35.0748599Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.0748599, -35.0748599}
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