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Simplifying 8a3 + -4a2 = a Reorder the terms: -4a2 + 8a3 = a Solving -4a2 + 8a3 = a Solving for variable 'a'. Reorder the terms: -1a + -4a2 + 8a3 = a + -1a Combine like terms: a + -1a = 0 -1a + -4a2 + 8a3 = 0 Factor out the Greatest Common Factor (GCF), 'a'. a(-1 + -4a + 8a2) = 0Subproblem 1
Set the factor 'a' equal to zero and attempt to solve: Simplifying a = 0 Solving a = 0 Move all terms containing a to the left, all other terms to the right. Simplifying a = 0Subproblem 2
Set the factor '(-1 + -4a + 8a2)' equal to zero and attempt to solve: Simplifying -1 + -4a + 8a2 = 0 Solving -1 + -4a + 8a2 = 0 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 = 0.683012702 a = 0.683012702 Simplifying a = 0.683012702Subproblem 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 = -0.183012702 a = -0.183012702 Simplifying a = -0.183012702Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.683012702, -0.183012702}Solution
a = {0, 0.683012702, -0.183012702}
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