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Simplifying 3a + -1[2a2 + -1(5a + -5a2 + 17)] = 0 Reorder the terms: 3a + -1[2a2 + -1(17 + 5a + -5a2)] = 0 3a + -1[2a2 + (17 * -1 + 5a * -1 + -5a2 * -1)] = 0 3a + -1[2a2 + (-17 + -5a + 5a2)] = 0 Reorder the terms: 3a + -1[-17 + -5a + 2a2 + 5a2] = 0 Combine like terms: 2a2 + 5a2 = 7a2 3a + -1[-17 + -5a + 7a2] = 0 3a + [-17 * -1 + -5a * -1 + 7a2 * -1] = 0 3a + [17 + 5a + -7a2] = 0 Reorder the terms: 17 + 3a + 5a + -7a2 = 0 Combine like terms: 3a + 5a = 8a 17 + 8a + -7a2 = 0 Solving 17 + 8a + -7a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -2.428571429 + -1.142857143a + a2 = 0 Move the constant term to the right: Add '2.428571429' to each side of the equation. -2.428571429 + -1.142857143a + 2.428571429 + a2 = 0 + 2.428571429 Reorder the terms: -2.428571429 + 2.428571429 + -1.142857143a + a2 = 0 + 2.428571429 Combine like terms: -2.428571429 + 2.428571429 = 0.000000000 0.000000000 + -1.142857143a + a2 = 0 + 2.428571429 -1.142857143a + a2 = 0 + 2.428571429 Combine like terms: 0 + 2.428571429 = 2.428571429 -1.142857143a + a2 = 2.428571429 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 = 2.428571429 + 0.3265306123 Reorder the terms: 0.3265306123 + -1.142857143a + a2 = 2.428571429 + 0.3265306123 Combine like terms: 2.428571429 + 0.3265306123 = 2.7551020413 0.3265306123 + -1.142857143a + a2 = 2.7551020413 Factor a perfect square on the left side: (a + -0.5714285715)(a + -0.5714285715) = 2.7551020413 Calculate the square root of the right side: 1.659850006 Break this problem into two subproblems by setting (a + -0.5714285715) equal to 1.659850006 and -1.659850006.Subproblem 1
a + -0.5714285715 = 1.659850006 Simplifying a + -0.5714285715 = 1.659850006 Reorder the terms: -0.5714285715 + a = 1.659850006 Solving -0.5714285715 + a = 1.659850006 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 = 1.659850006 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + a = 1.659850006 + 0.5714285715 a = 1.659850006 + 0.5714285715 Combine like terms: 1.659850006 + 0.5714285715 = 2.2312785775 a = 2.2312785775 Simplifying a = 2.2312785775Subproblem 2
a + -0.5714285715 = -1.659850006 Simplifying a + -0.5714285715 = -1.659850006 Reorder the terms: -0.5714285715 + a = -1.659850006 Solving -0.5714285715 + a = -1.659850006 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 = -1.659850006 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + a = -1.659850006 + 0.5714285715 a = -1.659850006 + 0.5714285715 Combine like terms: -1.659850006 + 0.5714285715 = -1.0884214345 a = -1.0884214345 Simplifying a = -1.0884214345Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.2312785775, -1.0884214345}
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