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Simplifying 9a(2a + -5) + 6a(3a + -2) = 1 Reorder the terms: 9a(-5 + 2a) + 6a(3a + -2) = 1 (-5 * 9a + 2a * 9a) + 6a(3a + -2) = 1 (-45a + 18a2) + 6a(3a + -2) = 1 Reorder the terms: -45a + 18a2 + 6a(-2 + 3a) = 1 -45a + 18a2 + (-2 * 6a + 3a * 6a) = 1 -45a + 18a2 + (-12a + 18a2) = 1 Reorder the terms: -45a + -12a + 18a2 + 18a2 = 1 Combine like terms: -45a + -12a = -57a -57a + 18a2 + 18a2 = 1 Combine like terms: 18a2 + 18a2 = 36a2 -57a + 36a2 = 1 Solving -57a + 36a2 = 1 Solving for variable 'a'. Reorder the terms: -1 + -57a + 36a2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -57a + 36a2 = 0 Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. -0.02777777778 + -1.583333333a + a2 = 0 Move the constant term to the right: Add '0.02777777778' to each side of the equation. -0.02777777778 + -1.583333333a + 0.02777777778 + a2 = 0 + 0.02777777778 Reorder the terms: -0.02777777778 + 0.02777777778 + -1.583333333a + a2 = 0 + 0.02777777778 Combine like terms: -0.02777777778 + 0.02777777778 = 0.00000000000 0.00000000000 + -1.583333333a + a2 = 0 + 0.02777777778 -1.583333333a + a2 = 0 + 0.02777777778 Combine like terms: 0 + 0.02777777778 = 0.02777777778 -1.583333333a + a2 = 0.02777777778 The a term is -1.583333333a. Take half its coefficient (-0.7916666665). Square it (0.6267361108) and add it to both sides. Add '0.6267361108' to each side of the equation. -1.583333333a + 0.6267361108 + a2 = 0.02777777778 + 0.6267361108 Reorder the terms: 0.6267361108 + -1.583333333a + a2 = 0.02777777778 + 0.6267361108 Combine like terms: 0.02777777778 + 0.6267361108 = 0.65451388858 0.6267361108 + -1.583333333a + a2 = 0.65451388858 Factor a perfect square on the left side: (a + -0.7916666665)(a + -0.7916666665) = 0.65451388858 Calculate the square root of the right side: 0.809020326 Break this problem into two subproblems by setting (a + -0.7916666665) equal to 0.809020326 and -0.809020326.Subproblem 1
a + -0.7916666665 = 0.809020326 Simplifying a + -0.7916666665 = 0.809020326 Reorder the terms: -0.7916666665 + a = 0.809020326 Solving -0.7916666665 + a = 0.809020326 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.7916666665' to each side of the equation. -0.7916666665 + 0.7916666665 + a = 0.809020326 + 0.7916666665 Combine like terms: -0.7916666665 + 0.7916666665 = 0.0000000000 0.0000000000 + a = 0.809020326 + 0.7916666665 a = 0.809020326 + 0.7916666665 Combine like terms: 0.809020326 + 0.7916666665 = 1.6006869925 a = 1.6006869925 Simplifying a = 1.6006869925Subproblem 2
a + -0.7916666665 = -0.809020326 Simplifying a + -0.7916666665 = -0.809020326 Reorder the terms: -0.7916666665 + a = -0.809020326 Solving -0.7916666665 + a = -0.809020326 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.7916666665' to each side of the equation. -0.7916666665 + 0.7916666665 + a = -0.809020326 + 0.7916666665 Combine like terms: -0.7916666665 + 0.7916666665 = 0.0000000000 0.0000000000 + a = -0.809020326 + 0.7916666665 a = -0.809020326 + 0.7916666665 Combine like terms: -0.809020326 + 0.7916666665 = -0.0173536595 a = -0.0173536595 Simplifying a = -0.0173536595Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.6006869925, -0.0173536595}
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