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Simplifying 9a2 = 12a + 1 Reorder the terms: 9a2 = 1 + 12a Solving 9a2 = 1 + 12a Solving for variable 'a'. Reorder the terms: -1 + -12a + 9a2 = 1 + 12a + -1 + -12a Reorder the terms: -1 + -12a + 9a2 = 1 + -1 + 12a + -12a Combine like terms: 1 + -1 = 0 -1 + -12a + 9a2 = 0 + 12a + -12a -1 + -12a + 9a2 = 12a + -12a Combine like terms: 12a + -12a = 0 -1 + -12a + 9a2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -1.333333333a + a2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -1.333333333a + 0.1111111111 + a2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -1.333333333a + a2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -1.333333333a + a2 = 0 + 0.1111111111 -1.333333333a + a2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -1.333333333a + a2 = 0.1111111111 The a term is -1.333333333a. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333a + 0.4444444442 + a2 = 0.1111111111 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333a + a2 = 0.1111111111 + 0.4444444442 Combine like terms: 0.1111111111 + 0.4444444442 = 0.5555555553 0.4444444442 + -1.333333333a + a2 = 0.5555555553 Factor a perfect square on the left side: (a + -0.6666666665)(a + -0.6666666665) = 0.5555555553 Calculate the square root of the right side: 0.745355992 Break this problem into two subproblems by setting (a + -0.6666666665) equal to 0.745355992 and -0.745355992.Subproblem 1
a + -0.6666666665 = 0.745355992 Simplifying a + -0.6666666665 = 0.745355992 Reorder the terms: -0.6666666665 + a = 0.745355992 Solving -0.6666666665 + a = 0.745355992 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + a = 0.745355992 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + a = 0.745355992 + 0.6666666665 a = 0.745355992 + 0.6666666665 Combine like terms: 0.745355992 + 0.6666666665 = 1.4120226585 a = 1.4120226585 Simplifying a = 1.4120226585Subproblem 2
a + -0.6666666665 = -0.745355992 Simplifying a + -0.6666666665 = -0.745355992 Reorder the terms: -0.6666666665 + a = -0.745355992 Solving -0.6666666665 + a = -0.745355992 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + a = -0.745355992 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + a = -0.745355992 + 0.6666666665 a = -0.745355992 + 0.6666666665 Combine like terms: -0.745355992 + 0.6666666665 = -0.0786893255 a = -0.0786893255 Simplifying a = -0.0786893255Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.4120226585, -0.0786893255}
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