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Simplifying 7a2 + 52a + 28 = 0 Reorder the terms: 28 + 52a + 7a2 = 0 Solving 28 + 52a + 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'. 4 + 7.428571429a + a2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + 7.428571429a + -4 + a2 = 0 + -4 Reorder the terms: 4 + -4 + 7.428571429a + a2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + 7.428571429a + a2 = 0 + -4 7.428571429a + a2 = 0 + -4 Combine like terms: 0 + -4 = -4 7.428571429a + a2 = -4 The a term is 7.428571429a. Take half its coefficient (3.714285715). Square it (13.79591837) and add it to both sides. Add '13.79591837' to each side of the equation. 7.428571429a + 13.79591837 + a2 = -4 + 13.79591837 Reorder the terms: 13.79591837 + 7.428571429a + a2 = -4 + 13.79591837 Combine like terms: -4 + 13.79591837 = 9.79591837 13.79591837 + 7.428571429a + a2 = 9.79591837 Factor a perfect square on the left side: (a + 3.714285715)(a + 3.714285715) = 9.79591837 Calculate the square root of the right side: 3.129843186 Break this problem into two subproblems by setting (a + 3.714285715) equal to 3.129843186 and -3.129843186.Subproblem 1
a + 3.714285715 = 3.129843186 Simplifying a + 3.714285715 = 3.129843186 Reorder the terms: 3.714285715 + a = 3.129843186 Solving 3.714285715 + a = 3.129843186 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.714285715' to each side of the equation. 3.714285715 + -3.714285715 + a = 3.129843186 + -3.714285715 Combine like terms: 3.714285715 + -3.714285715 = 0.000000000 0.000000000 + a = 3.129843186 + -3.714285715 a = 3.129843186 + -3.714285715 Combine like terms: 3.129843186 + -3.714285715 = -0.584442529 a = -0.584442529 Simplifying a = -0.584442529Subproblem 2
a + 3.714285715 = -3.129843186 Simplifying a + 3.714285715 = -3.129843186 Reorder the terms: 3.714285715 + a = -3.129843186 Solving 3.714285715 + a = -3.129843186 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-3.714285715' to each side of the equation. 3.714285715 + -3.714285715 + a = -3.129843186 + -3.714285715 Combine like terms: 3.714285715 + -3.714285715 = 0.000000000 0.000000000 + a = -3.129843186 + -3.714285715 a = -3.129843186 + -3.714285715 Combine like terms: -3.129843186 + -3.714285715 = -6.844128901 a = -6.844128901 Simplifying a = -6.844128901Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.584442529, -6.844128901}
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