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Simplifying 6a(1 + 8) * a = 9a + 1 Combine like terms: 1 + 8 = 9 6a(9) * a = 9a + 1 Reorder the terms for easier multiplication: 6 * 9a * a = 9a + 1 Multiply 6 * 9 54a * a = 9a + 1 Multiply a * a 54a2 = 9a + 1 Reorder the terms: 54a2 = 1 + 9a Solving 54a2 = 1 + 9a Solving for variable 'a'. Reorder the terms: -1 + -9a + 54a2 = 1 + 9a + -1 + -9a Reorder the terms: -1 + -9a + 54a2 = 1 + -1 + 9a + -9a Combine like terms: 1 + -1 = 0 -1 + -9a + 54a2 = 0 + 9a + -9a -1 + -9a + 54a2 = 9a + -9a Combine like terms: 9a + -9a = 0 -1 + -9a + 54a2 = 0 Begin completing the square. Divide all terms by 54 the coefficient of the squared term: Divide each side by '54'. -0.01851851852 + -0.1666666667a + a2 = 0 Move the constant term to the right: Add '0.01851851852' to each side of the equation. -0.01851851852 + -0.1666666667a + 0.01851851852 + a2 = 0 + 0.01851851852 Reorder the terms: -0.01851851852 + 0.01851851852 + -0.1666666667a + a2 = 0 + 0.01851851852 Combine like terms: -0.01851851852 + 0.01851851852 = 0.00000000000 0.00000000000 + -0.1666666667a + a2 = 0 + 0.01851851852 -0.1666666667a + a2 = 0 + 0.01851851852 Combine like terms: 0 + 0.01851851852 = 0.01851851852 -0.1666666667a + a2 = 0.01851851852 The a term is -0.1666666667a. Take half its coefficient (-0.08333333335). Square it (0.006944444447) and add it to both sides. Add '0.006944444447' to each side of the equation. -0.1666666667a + 0.006944444447 + a2 = 0.01851851852 + 0.006944444447 Reorder the terms: 0.006944444447 + -0.1666666667a + a2 = 0.01851851852 + 0.006944444447 Combine like terms: 0.01851851852 + 0.006944444447 = 0.025462962967 0.006944444447 + -0.1666666667a + a2 = 0.025462962967 Factor a perfect square on the left side: (a + -0.08333333335)(a + -0.08333333335) = 0.025462962967 Calculate the square root of the right side: 0.159571185 Break this problem into two subproblems by setting (a + -0.08333333335) equal to 0.159571185 and -0.159571185.Subproblem 1
a + -0.08333333335 = 0.159571185 Simplifying a + -0.08333333335 = 0.159571185 Reorder the terms: -0.08333333335 + a = 0.159571185 Solving -0.08333333335 + a = 0.159571185 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.08333333335' to each side of the equation. -0.08333333335 + 0.08333333335 + a = 0.159571185 + 0.08333333335 Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000 0.00000000000 + a = 0.159571185 + 0.08333333335 a = 0.159571185 + 0.08333333335 Combine like terms: 0.159571185 + 0.08333333335 = 0.24290451835 a = 0.24290451835 Simplifying a = 0.24290451835Subproblem 2
a + -0.08333333335 = -0.159571185 Simplifying a + -0.08333333335 = -0.159571185 Reorder the terms: -0.08333333335 + a = -0.159571185 Solving -0.08333333335 + a = -0.159571185 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.08333333335' to each side of the equation. -0.08333333335 + 0.08333333335 + a = -0.159571185 + 0.08333333335 Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000 0.00000000000 + a = -0.159571185 + 0.08333333335 a = -0.159571185 + 0.08333333335 Combine like terms: -0.159571185 + 0.08333333335 = -0.07623785165 a = -0.07623785165 Simplifying a = -0.07623785165Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.24290451835, -0.07623785165}
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