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Simplifying 6(2a + -1) * 6a = 12 Reorder the terms: 6(-1 + 2a) * 6a = 12 Reorder the terms for easier multiplication: 6 * 6a(-1 + 2a) = 12 Multiply 6 * 6 36a(-1 + 2a) = 12 (-1 * 36a + 2a * 36a) = 12 (-36a + 72a2) = 12 Solving -36a + 72a2 = 12 Solving for variable 'a'. Reorder the terms: -12 + -36a + 72a2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + -36a + 72a2 = 0 Factor out the Greatest Common Factor (GCF), '12'. 12(-1 + -3a + 6a2) = 0 Ignore the factor 12.Subproblem 1
Set the factor '(-1 + -3a + 6a2)' equal to zero and attempt to solve: Simplifying -1 + -3a + 6a2 = 0 Solving -1 + -3a + 6a2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.1666666667 + -0.5a + a2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + -0.5a + 0.1666666667 + a2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + -0.5a + a2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + -0.5a + a2 = 0 + 0.1666666667 -0.5a + a2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 -0.5a + a2 = 0.1666666667 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 = 0.1666666667 + 0.0625 Reorder the terms: 0.0625 + -0.5a + a2 = 0.1666666667 + 0.0625 Combine like terms: 0.1666666667 + 0.0625 = 0.2291666667 0.0625 + -0.5a + a2 = 0.2291666667 Factor a perfect square on the left side: (a + -0.25)(a + -0.25) = 0.2291666667 Calculate the square root of the right side: 0.478713554 Break this problem into two subproblems by setting (a + -0.25) equal to 0.478713554 and -0.478713554.Subproblem 1
a + -0.25 = 0.478713554 Simplifying a + -0.25 = 0.478713554 Reorder the terms: -0.25 + a = 0.478713554 Solving -0.25 + a = 0.478713554 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 = 0.478713554 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 0.478713554 + 0.25 a = 0.478713554 + 0.25 Combine like terms: 0.478713554 + 0.25 = 0.728713554 a = 0.728713554 Simplifying a = 0.728713554Subproblem 2
a + -0.25 = -0.478713554 Simplifying a + -0.25 = -0.478713554 Reorder the terms: -0.25 + a = -0.478713554 Solving -0.25 + a = -0.478713554 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 = -0.478713554 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -0.478713554 + 0.25 a = -0.478713554 + 0.25 Combine like terms: -0.478713554 + 0.25 = -0.228713554 a = -0.228713554 Simplifying a = -0.228713554Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.728713554, -0.228713554}Solution
a = {0.728713554, -0.228713554}
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