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Simplifying (6a2 + 8) + (2a2 + 4a + -9) = 0 Reorder the terms: (8 + 6a2) + (2a2 + 4a + -9) = 0 Remove parenthesis around (8 + 6a2) 8 + 6a2 + (2a2 + 4a + -9) = 0 Reorder the terms: 8 + 6a2 + (-9 + 4a + 2a2) = 0 Remove parenthesis around (-9 + 4a + 2a2) 8 + 6a2 + -9 + 4a + 2a2 = 0 Reorder the terms: 8 + -9 + 4a + 6a2 + 2a2 = 0 Combine like terms: 8 + -9 = -1 -1 + 4a + 6a2 + 2a2 = 0 Combine like terms: 6a2 + 2a2 = 8a2 -1 + 4a + 8a2 = 0 Solving -1 + 4a + 8a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + 0.5a + a2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 0.5a + 0.125 + a2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 0.5a + a2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 0.5a + a2 = 0 + 0.125 0.5a + a2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 0.5a + a2 = 0.125 The a term is 0.5a. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5a + 0.0625 + a2 = 0.125 + 0.0625 Reorder the terms: 0.0625 + 0.5a + a2 = 0.125 + 0.0625 Combine like terms: 0.125 + 0.0625 = 0.1875 0.0625 + 0.5a + a2 = 0.1875 Factor a perfect square on the left side: (a + 0.25)(a + 0.25) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (a + 0.25) equal to 0.433012702 and -0.433012702.Subproblem 1
a + 0.25 = 0.433012702 Simplifying a + 0.25 = 0.433012702 Reorder the terms: 0.25 + a = 0.433012702 Solving 0.25 + a = 0.433012702 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = 0.433012702 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = 0.433012702 + -0.25 a = 0.433012702 + -0.25 Combine like terms: 0.433012702 + -0.25 = 0.183012702 a = 0.183012702 Simplifying a = 0.183012702Subproblem 2
a + 0.25 = -0.433012702 Simplifying a + 0.25 = -0.433012702 Reorder the terms: 0.25 + a = -0.433012702 Solving 0.25 + a = -0.433012702 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = -0.433012702 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = -0.433012702 + -0.25 a = -0.433012702 + -0.25 Combine like terms: -0.433012702 + -0.25 = -0.683012702 a = -0.683012702 Simplifying a = -0.683012702Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.183012702, -0.683012702}
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