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Simplifying 3a2 + -21a + 33 = 0 Reorder the terms: 33 + -21a + 3a2 = 0 Solving 33 + -21a + 3a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '3'. 3(11 + -7a + a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(11 + -7a + a2)' equal to zero and attempt to solve: Simplifying 11 + -7a + a2 = 0 Solving 11 + -7a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '-11' to each side of the equation. 11 + -7a + -11 + a2 = 0 + -11 Reorder the terms: 11 + -11 + -7a + a2 = 0 + -11 Combine like terms: 11 + -11 = 0 0 + -7a + a2 = 0 + -11 -7a + a2 = 0 + -11 Combine like terms: 0 + -11 = -11 -7a + a2 = -11 The a term is -7a. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7a + 12.25 + a2 = -11 + 12.25 Reorder the terms: 12.25 + -7a + a2 = -11 + 12.25 Combine like terms: -11 + 12.25 = 1.25 12.25 + -7a + a2 = 1.25 Factor a perfect square on the left side: (a + -3.5)(a + -3.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (a + -3.5) equal to 1.118033989 and -1.118033989.Subproblem 1
a + -3.5 = 1.118033989 Simplifying a + -3.5 = 1.118033989 Reorder the terms: -3.5 + a = 1.118033989 Solving -3.5 + a = 1.118033989 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + a = 1.118033989 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + a = 1.118033989 + 3.5 a = 1.118033989 + 3.5 Combine like terms: 1.118033989 + 3.5 = 4.618033989 a = 4.618033989 Simplifying a = 4.618033989Subproblem 2
a + -3.5 = -1.118033989 Simplifying a + -3.5 = -1.118033989 Reorder the terms: -3.5 + a = -1.118033989 Solving -3.5 + a = -1.118033989 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + a = -1.118033989 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + a = -1.118033989 + 3.5 a = -1.118033989 + 3.5 Combine like terms: -1.118033989 + 3.5 = 2.381966011 a = 2.381966011 Simplifying a = 2.381966011Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.618033989, 2.381966011}Solution
a = {4.618033989, 2.381966011}
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