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Simplifying 6a(7a2 + 8a + -5) = 0 Reorder the terms: 6a(-5 + 8a + 7a2) = 0 (-5 * 6a + 8a * 6a + 7a2 * 6a) = 0 (-30a + 48a2 + 42a3) = 0 Solving -30a + 48a2 + 42a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '6a'. 6a(-5 + 8a + 7a2) = 0 Ignore the factor 6.Subproblem 1
Set the factor 'a' equal to zero and attempt to solve: Simplifying a = 0 Solving a = 0 Move all terms containing a to the left, all other terms to the right. Simplifying a = 0Subproblem 2
Set the factor '(-5 + 8a + 7a2)' equal to zero and attempt to solve: Simplifying -5 + 8a + 7a2 = 0 Solving -5 + 8a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.7142857143 + 1.142857143a + a2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + 1.142857143a + 0.7142857143 + a2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + 1.142857143a + a2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + 1.142857143a + a2 = 0 + 0.7142857143 1.142857143a + a2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 1.142857143a + a2 = 0.7142857143 The a term is 1.142857143a. Take half its coefficient (0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. 1.142857143a + 0.3265306123 + a2 = 0.7142857143 + 0.3265306123 Reorder the terms: 0.3265306123 + 1.142857143a + a2 = 0.7142857143 + 0.3265306123 Combine like terms: 0.7142857143 + 0.3265306123 = 1.0408163266 0.3265306123 + 1.142857143a + a2 = 1.0408163266 Factor a perfect square on the left side: (a + 0.5714285715)(a + 0.5714285715) = 1.0408163266 Calculate the square root of the right side: 1.020204061 Break this problem into two subproblems by setting (a + 0.5714285715) equal to 1.020204061 and -1.020204061.Subproblem 1
a + 0.5714285715 = 1.020204061 Simplifying a + 0.5714285715 = 1.020204061 Reorder the terms: 0.5714285715 + a = 1.020204061 Solving 0.5714285715 + a = 1.020204061 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + a = 1.020204061 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + a = 1.020204061 + -0.5714285715 a = 1.020204061 + -0.5714285715 Combine like terms: 1.020204061 + -0.5714285715 = 0.4487754895 a = 0.4487754895 Simplifying a = 0.4487754895Subproblem 2
a + 0.5714285715 = -1.020204061 Simplifying a + 0.5714285715 = -1.020204061 Reorder the terms: 0.5714285715 + a = -1.020204061 Solving 0.5714285715 + a = -1.020204061 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + a = -1.020204061 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + a = -1.020204061 + -0.5714285715 a = -1.020204061 + -0.5714285715 Combine like terms: -1.020204061 + -0.5714285715 = -1.5916326325 a = -1.5916326325 Simplifying a = -1.5916326325Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.4487754895, -1.5916326325}Solution
a = {0, 0.4487754895, -1.5916326325}
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