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Simplifying 4a2 + -4a + -14 = 0 Reorder the terms: -14 + -4a + 4a2 = 0 Solving -14 + -4a + 4a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -2a + 2a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -2a + 2a2)' equal to zero and attempt to solve: Simplifying -7 + -2a + 2a2 = 0 Solving -7 + -2a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + -1a + a2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -1a + 3.5 + a2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -1a + a2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -1a + a2 = 0 + 3.5 -1a + a2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -1a + a2 = 3.5 The a term is -1a. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1a + 0.25 + a2 = 3.5 + 0.25 Reorder the terms: 0.25 + -1a + a2 = 3.5 + 0.25 Combine like terms: 3.5 + 0.25 = 3.75 0.25 + -1a + a2 = 3.75 Factor a perfect square on the left side: (a + -0.5)(a + -0.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (a + -0.5) equal to 1.936491673 and -1.936491673.Subproblem 1
a + -0.5 = 1.936491673 Simplifying a + -0.5 = 1.936491673 Reorder the terms: -0.5 + a = 1.936491673 Solving -0.5 + a = 1.936491673 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = 1.936491673 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = 1.936491673 + 0.5 a = 1.936491673 + 0.5 Combine like terms: 1.936491673 + 0.5 = 2.436491673 a = 2.436491673 Simplifying a = 2.436491673Subproblem 2
a + -0.5 = -1.936491673 Simplifying a + -0.5 = -1.936491673 Reorder the terms: -0.5 + a = -1.936491673 Solving -0.5 + a = -1.936491673 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = -1.936491673 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = -1.936491673 + 0.5 a = -1.936491673 + 0.5 Combine like terms: -1.936491673 + 0.5 = -1.436491673 a = -1.436491673 Simplifying a = -1.436491673Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.436491673, -1.436491673}Solution
a = {2.436491673, -1.436491673}
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