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Simplifying 30a2 + 174a + 174 = 0 Reorder the terms: 174 + 174a + 30a2 = 0 Solving 174 + 174a + 30a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '6'. 6(29 + 29a + 5a2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(29 + 29a + 5a2)' equal to zero and attempt to solve: Simplifying 29 + 29a + 5a2 = 0 Solving 29 + 29a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 5.8 + 5.8a + a2 = 0 Move the constant term to the right: Add '-5.8' to each side of the equation. 5.8 + 5.8a + -5.8 + a2 = 0 + -5.8 Reorder the terms: 5.8 + -5.8 + 5.8a + a2 = 0 + -5.8 Combine like terms: 5.8 + -5.8 = 0.0 0.0 + 5.8a + a2 = 0 + -5.8 5.8a + a2 = 0 + -5.8 Combine like terms: 0 + -5.8 = -5.8 5.8a + a2 = -5.8 The a term is 5.8a. Take half its coefficient (2.9). Square it (8.41) and add it to both sides. Add '8.41' to each side of the equation. 5.8a + 8.41 + a2 = -5.8 + 8.41 Reorder the terms: 8.41 + 5.8a + a2 = -5.8 + 8.41 Combine like terms: -5.8 + 8.41 = 2.61 8.41 + 5.8a + a2 = 2.61 Factor a perfect square on the left side: (a + 2.9)(a + 2.9) = 2.61 Calculate the square root of the right side: 1.615549442 Break this problem into two subproblems by setting (a + 2.9) equal to 1.615549442 and -1.615549442.Subproblem 1
a + 2.9 = 1.615549442 Simplifying a + 2.9 = 1.615549442 Reorder the terms: 2.9 + a = 1.615549442 Solving 2.9 + a = 1.615549442 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.9' to each side of the equation. 2.9 + -2.9 + a = 1.615549442 + -2.9 Combine like terms: 2.9 + -2.9 = 0.0 0.0 + a = 1.615549442 + -2.9 a = 1.615549442 + -2.9 Combine like terms: 1.615549442 + -2.9 = -1.284450558 a = -1.284450558 Simplifying a = -1.284450558Subproblem 2
a + 2.9 = -1.615549442 Simplifying a + 2.9 = -1.615549442 Reorder the terms: 2.9 + a = -1.615549442 Solving 2.9 + a = -1.615549442 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.9' to each side of the equation. 2.9 + -2.9 + a = -1.615549442 + -2.9 Combine like terms: 2.9 + -2.9 = 0.0 0.0 + a = -1.615549442 + -2.9 a = -1.615549442 + -2.9 Combine like terms: -1.615549442 + -2.9 = -4.515549442 a = -4.515549442 Simplifying a = -4.515549442Solution
The solution to the problem is based on the solutions from the subproblems. a = {-1.284450558, -4.515549442}Solution
a = {-1.284450558, -4.515549442}
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