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Simplifying -4a(8a + 17) + 7(9 + 4) = 0 Reorder the terms: -4a(17 + 8a) + 7(9 + 4) = 0 (17 * -4a + 8a * -4a) + 7(9 + 4) = 0 (-68a + -32a2) + 7(9 + 4) = 0 Combine like terms: 9 + 4 = 13 -68a + -32a2 + 7(13) = 0 Multiply 7 * 13 -68a + -32a2 + 91 = 0 Reorder the terms: 91 + -68a + -32a2 = 0 Solving 91 + -68a + -32a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -32 the coefficient of the squared term: Divide each side by '-32'. -2.84375 + 2.125a + a2 = 0 Move the constant term to the right: Add '2.84375' to each side of the equation. -2.84375 + 2.125a + 2.84375 + a2 = 0 + 2.84375 Reorder the terms: -2.84375 + 2.84375 + 2.125a + a2 = 0 + 2.84375 Combine like terms: -2.84375 + 2.84375 = 0.00000 0.00000 + 2.125a + a2 = 0 + 2.84375 2.125a + a2 = 0 + 2.84375 Combine like terms: 0 + 2.84375 = 2.84375 2.125a + a2 = 2.84375 The a term is 2.125a. Take half its coefficient (1.0625). Square it (1.12890625) and add it to both sides. Add '1.12890625' to each side of the equation. 2.125a + 1.12890625 + a2 = 2.84375 + 1.12890625 Reorder the terms: 1.12890625 + 2.125a + a2 = 2.84375 + 1.12890625 Combine like terms: 2.84375 + 1.12890625 = 3.97265625 1.12890625 + 2.125a + a2 = 3.97265625 Factor a perfect square on the left side: (a + 1.0625)(a + 1.0625) = 3.97265625 Calculate the square root of the right side: 1.99315234 Break this problem into two subproblems by setting (a + 1.0625) equal to 1.99315234 and -1.99315234.Subproblem 1
a + 1.0625 = 1.99315234 Simplifying a + 1.0625 = 1.99315234 Reorder the terms: 1.0625 + a = 1.99315234 Solving 1.0625 + a = 1.99315234 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.0625' to each side of the equation. 1.0625 + -1.0625 + a = 1.99315234 + -1.0625 Combine like terms: 1.0625 + -1.0625 = 0.0000 0.0000 + a = 1.99315234 + -1.0625 a = 1.99315234 + -1.0625 Combine like terms: 1.99315234 + -1.0625 = 0.93065234 a = 0.93065234 Simplifying a = 0.93065234Subproblem 2
a + 1.0625 = -1.99315234 Simplifying a + 1.0625 = -1.99315234 Reorder the terms: 1.0625 + a = -1.99315234 Solving 1.0625 + a = -1.99315234 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.0625' to each side of the equation. 1.0625 + -1.0625 + a = -1.99315234 + -1.0625 Combine like terms: 1.0625 + -1.0625 = 0.0000 0.0000 + a = -1.99315234 + -1.0625 a = -1.99315234 + -1.0625 Combine like terms: -1.99315234 + -1.0625 = -3.05565234 a = -3.05565234 Simplifying a = -3.05565234Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.93065234, -3.05565234}
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