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Simplifying 7a2 + 17a + 9 = 0 Reorder the terms: 9 + 17a + 7a2 = 0 Solving 9 + 17a + 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'. 1.285714286 + 2.428571429a + a2 = 0 Move the constant term to the right: Add '-1.285714286' to each side of the equation. 1.285714286 + 2.428571429a + -1.285714286 + a2 = 0 + -1.285714286 Reorder the terms: 1.285714286 + -1.285714286 + 2.428571429a + a2 = 0 + -1.285714286 Combine like terms: 1.285714286 + -1.285714286 = 0.000000000 0.000000000 + 2.428571429a + a2 = 0 + -1.285714286 2.428571429a + a2 = 0 + -1.285714286 Combine like terms: 0 + -1.285714286 = -1.285714286 2.428571429a + a2 = -1.285714286 The a term is 2.428571429a. Take half its coefficient (1.214285715). Square it (1.474489798) and add it to both sides. Add '1.474489798' to each side of the equation. 2.428571429a + 1.474489798 + a2 = -1.285714286 + 1.474489798 Reorder the terms: 1.474489798 + 2.428571429a + a2 = -1.285714286 + 1.474489798 Combine like terms: -1.285714286 + 1.474489798 = 0.188775512 1.474489798 + 2.428571429a + a2 = 0.188775512 Factor a perfect square on the left side: (a + 1.214285715)(a + 1.214285715) = 0.188775512 Calculate the square root of the right side: 0.43448304 Break this problem into two subproblems by setting (a + 1.214285715) equal to 0.43448304 and -0.43448304.Subproblem 1
a + 1.214285715 = 0.43448304 Simplifying a + 1.214285715 = 0.43448304 Reorder the terms: 1.214285715 + a = 0.43448304 Solving 1.214285715 + a = 0.43448304 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.214285715' to each side of the equation. 1.214285715 + -1.214285715 + a = 0.43448304 + -1.214285715 Combine like terms: 1.214285715 + -1.214285715 = 0.000000000 0.000000000 + a = 0.43448304 + -1.214285715 a = 0.43448304 + -1.214285715 Combine like terms: 0.43448304 + -1.214285715 = -0.779802675 a = -0.779802675 Simplifying a = -0.779802675Subproblem 2
a + 1.214285715 = -0.43448304 Simplifying a + 1.214285715 = -0.43448304 Reorder the terms: 1.214285715 + a = -0.43448304 Solving 1.214285715 + a = -0.43448304 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.214285715' to each side of the equation. 1.214285715 + -1.214285715 + a = -0.43448304 + -1.214285715 Combine like terms: 1.214285715 + -1.214285715 = 0.000000000 0.000000000 + a = -0.43448304 + -1.214285715 a = -0.43448304 + -1.214285715 Combine like terms: -0.43448304 + -1.214285715 = -1.648768755 a = -1.648768755 Simplifying a = -1.648768755Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.779802675, -1.648768755}
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