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Simplifying a2 + 13a + 22 = 7 Reorder the terms: 22 + 13a + a2 = 7 Solving 22 + 13a + a2 = 7 Solving for variable 'a'. Reorder the terms: 22 + -7 + 13a + a2 = 7 + -7 Combine like terms: 22 + -7 = 15 15 + 13a + a2 = 7 + -7 Combine like terms: 7 + -7 = 0 15 + 13a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-15' to each side of the equation. 15 + 13a + -15 + a2 = 0 + -15 Reorder the terms: 15 + -15 + 13a + a2 = 0 + -15 Combine like terms: 15 + -15 = 0 0 + 13a + a2 = 0 + -15 13a + a2 = 0 + -15 Combine like terms: 0 + -15 = -15 13a + a2 = -15 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 = -15 + 42.25 Reorder the terms: 42.25 + 13a + a2 = -15 + 42.25 Combine like terms: -15 + 42.25 = 27.25 42.25 + 13a + a2 = 27.25 Factor a perfect square on the left side: (a + 6.5)(a + 6.5) = 27.25 Calculate the square root of the right side: 5.220153254 Break this problem into two subproblems by setting (a + 6.5) equal to 5.220153254 and -5.220153254.Subproblem 1
a + 6.5 = 5.220153254 Simplifying a + 6.5 = 5.220153254 Reorder the terms: 6.5 + a = 5.220153254 Solving 6.5 + a = 5.220153254 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 = 5.220153254 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = 5.220153254 + -6.5 a = 5.220153254 + -6.5 Combine like terms: 5.220153254 + -6.5 = -1.279846746 a = -1.279846746 Simplifying a = -1.279846746Subproblem 2
a + 6.5 = -5.220153254 Simplifying a + 6.5 = -5.220153254 Reorder the terms: 6.5 + a = -5.220153254 Solving 6.5 + a = -5.220153254 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 = -5.220153254 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + a = -5.220153254 + -6.5 a = -5.220153254 + -6.5 Combine like terms: -5.220153254 + -6.5 = -11.720153254 a = -11.720153254 Simplifying a = -11.720153254Solution
The solution to the problem is based on the solutions from the subproblems. a = {-1.279846746, -11.720153254}
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