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Simplifying -12a2 + 23a + 20 = 0 Reorder the terms: 20 + 23a + -12a2 = 0 Solving 20 + 23a + -12a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -1.666666667 + -1.916666667a + a2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + -1.916666667a + 1.666666667 + a2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + -1.916666667a + a2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + -1.916666667a + a2 = 0 + 1.666666667 -1.916666667a + a2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 -1.916666667a + a2 = 1.666666667 The a term is -1.916666667a. Take half its coefficient (-0.9583333335). Square it (0.9184027781) and add it to both sides. Add '0.9184027781' to each side of the equation. -1.916666667a + 0.9184027781 + a2 = 1.666666667 + 0.9184027781 Reorder the terms: 0.9184027781 + -1.916666667a + a2 = 1.666666667 + 0.9184027781 Combine like terms: 1.666666667 + 0.9184027781 = 2.5850694451 0.9184027781 + -1.916666667a + a2 = 2.5850694451 Factor a perfect square on the left side: (a + -0.9583333335)(a + -0.9583333335) = 2.5850694451 Calculate the square root of the right side: 1.607815115 Break this problem into two subproblems by setting (a + -0.9583333335) equal to 1.607815115 and -1.607815115.Subproblem 1
a + -0.9583333335 = 1.607815115 Simplifying a + -0.9583333335 = 1.607815115 Reorder the terms: -0.9583333335 + a = 1.607815115 Solving -0.9583333335 + a = 1.607815115 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.9583333335' to each side of the equation. -0.9583333335 + 0.9583333335 + a = 1.607815115 + 0.9583333335 Combine like terms: -0.9583333335 + 0.9583333335 = 0.0000000000 0.0000000000 + a = 1.607815115 + 0.9583333335 a = 1.607815115 + 0.9583333335 Combine like terms: 1.607815115 + 0.9583333335 = 2.5661484485 a = 2.5661484485 Simplifying a = 2.5661484485Subproblem 2
a + -0.9583333335 = -1.607815115 Simplifying a + -0.9583333335 = -1.607815115 Reorder the terms: -0.9583333335 + a = -1.607815115 Solving -0.9583333335 + a = -1.607815115 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.9583333335' to each side of the equation. -0.9583333335 + 0.9583333335 + a = -1.607815115 + 0.9583333335 Combine like terms: -0.9583333335 + 0.9583333335 = 0.0000000000 0.0000000000 + a = -1.607815115 + 0.9583333335 a = -1.607815115 + 0.9583333335 Combine like terms: -1.607815115 + 0.9583333335 = -0.6494817815 a = -0.6494817815 Simplifying a = -0.6494817815Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.5661484485, -0.6494817815}
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