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Simplifying a2 + 2a = 32223 Reorder the terms: 2a + a2 = 32223 Solving 2a + a2 = 32223 Solving for variable 'a'. Reorder the terms: -32223 + 2a + a2 = 32223 + -32223 Combine like terms: 32223 + -32223 = 0 -32223 + 2a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '32223' to each side of the equation. -32223 + 2a + 32223 + a2 = 0 + 32223 Reorder the terms: -32223 + 32223 + 2a + a2 = 0 + 32223 Combine like terms: -32223 + 32223 = 0 0 + 2a + a2 = 0 + 32223 2a + a2 = 0 + 32223 Combine like terms: 0 + 32223 = 32223 2a + a2 = 32223 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 = 32223 + 1 Reorder the terms: 1 + 2a + a2 = 32223 + 1 Combine like terms: 32223 + 1 = 32224 1 + 2a + a2 = 32224 Factor a perfect square on the left side: (a + 1)(a + 1) = 32224 Calculate the square root of the right side: 179.510445379 Break this problem into two subproblems by setting (a + 1) equal to 179.510445379 and -179.510445379.Subproblem 1
a + 1 = 179.510445379 Simplifying a + 1 = 179.510445379 Reorder the terms: 1 + a = 179.510445379 Solving 1 + a = 179.510445379 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 = 179.510445379 + -1 Combine like terms: 1 + -1 = 0 0 + a = 179.510445379 + -1 a = 179.510445379 + -1 Combine like terms: 179.510445379 + -1 = 178.510445379 a = 178.510445379 Simplifying a = 178.510445379Subproblem 2
a + 1 = -179.510445379 Simplifying a + 1 = -179.510445379 Reorder the terms: 1 + a = -179.510445379 Solving 1 + a = -179.510445379 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 = -179.510445379 + -1 Combine like terms: 1 + -1 = 0 0 + a = -179.510445379 + -1 a = -179.510445379 + -1 Combine like terms: -179.510445379 + -1 = -180.510445379 a = -180.510445379 Simplifying a = -180.510445379Solution
The solution to the problem is based on the solutions from the subproblems. a = {178.510445379, -180.510445379}
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