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Simplifying a2 + 6a + 7a = -5 Reorder the terms: 6a + 7a + a2 = -5 Combine like terms: 6a + 7a = 13a 13a + a2 = -5 Solving 13a + a2 = -5 Solving for variable 'a'. Reorder the terms: 5 + 13a + a2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + 13a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 13a + -5 + a2 = 0 + -5 Reorder the terms: 5 + -5 + 13a + a2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 13a + a2 = 0 + -5 13a + a2 = 0 + -5 Combine like terms: 0 + -5 = -5 13a + a2 = -5 The a term is 13a. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13a + 42.25 + a2 = -5 + 42.25 Reorder the terms: 42.25 + 13a + a2 = -5 + 42.25 Combine like terms: -5 + 42.25 = 37.25 42.25 + 13a + a2 = 37.25 Factor a perfect square on the left side: (a + 6.5)(a + 6.5) = 37.25 Calculate the square root of the right side: 6.103277808 Break this problem into two subproblems by setting (a + 6.5) equal to 6.103277808 and -6.103277808.Subproblem 1
a + 6.5 = 6.103277808 Simplifying a + 6.5 = 6.103277808 Reorder the terms: 6.5 + a = 6.103277808 Solving 6.5 + a = 6.103277808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + a = 6.103277808 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = 6.103277808 + -6.5 a = 6.103277808 + -6.5 Combine like terms: 6.103277808 + -6.5 = -0.396722192 a = -0.396722192 Simplifying a = -0.396722192Subproblem 2
a + 6.5 = -6.103277808 Simplifying a + 6.5 = -6.103277808 Reorder the terms: 6.5 + a = -6.103277808 Solving 6.5 + a = -6.103277808 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + a = -6.103277808 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = -6.103277808 + -6.5 a = -6.103277808 + -6.5 Combine like terms: -6.103277808 + -6.5 = -12.603277808 a = -12.603277808 Simplifying a = -12.603277808Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.396722192, -12.603277808}
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