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Simplifying 5 + 2a + a2 + -12 = 0 Reorder the terms: 5 + -12 + 2a + a2 = 0 Combine like terms: 5 + -12 = -7 -7 + 2a + a2 = 0 Solving -7 + 2a + a2 = 0 Solving for variable 'a'. Begin completing the square. Move the constant term to the right: Add '7' to each side of the equation. -7 + 2a + 7 + a2 = 0 + 7 Reorder the terms: -7 + 7 + 2a + a2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 2a + a2 = 0 + 7 2a + a2 = 0 + 7 Combine like terms: 0 + 7 = 7 2a + a2 = 7 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 = 7 + 1 Reorder the terms: 1 + 2a + a2 = 7 + 1 Combine like terms: 7 + 1 = 8 1 + 2a + a2 = 8 Factor a perfect square on the left side: (a + 1)(a + 1) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (a + 1) equal to 2.828427125 and -2.828427125.Subproblem 1
a + 1 = 2.828427125 Simplifying a + 1 = 2.828427125 Reorder the terms: 1 + a = 2.828427125 Solving 1 + a = 2.828427125 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 = 2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + a = 2.828427125 + -1 a = 2.828427125 + -1 Combine like terms: 2.828427125 + -1 = 1.828427125 a = 1.828427125 Simplifying a = 1.828427125Subproblem 2
a + 1 = -2.828427125 Simplifying a + 1 = -2.828427125 Reorder the terms: 1 + a = -2.828427125 Solving 1 + a = -2.828427125 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 = -2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + a = -2.828427125 + -1 a = -2.828427125 + -1 Combine like terms: -2.828427125 + -1 = -3.828427125 a = -3.828427125 Simplifying a = -3.828427125Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.828427125, -3.828427125}
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