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Simplifying (7a2 + -6a + 3) + -1(a2 + -7a + 9) = 0 Reorder the terms: (3 + -6a + 7a2) + -1(a2 + -7a + 9) = 0 Remove parenthesis around (3 + -6a + 7a2) 3 + -6a + 7a2 + -1(a2 + -7a + 9) = 0 Reorder the terms: 3 + -6a + 7a2 + -1(9 + -7a + a2) = 0 3 + -6a + 7a2 + (9 * -1 + -7a * -1 + a2 * -1) = 0 3 + -6a + 7a2 + (-9 + 7a + -1a2) = 0 Reorder the terms: 3 + -9 + -6a + 7a + 7a2 + -1a2 = 0 Combine like terms: 3 + -9 = -6 -6 + -6a + 7a + 7a2 + -1a2 = 0 Combine like terms: -6a + 7a = 1a -6 + 1a + 7a2 + -1a2 = 0 Combine like terms: 7a2 + -1a2 = 6a2 -6 + 1a + 6a2 = 0 Solving -6 + 1a + 6a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1 + 0.1666666667a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.1666666667a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + 0.1666666667a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.1666666667a + a2 = 0 + 1 0.1666666667a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.1666666667a + a2 = 1 The a term is 0.1666666667a. Take half its coefficient (0.08333333335). Square it (0.006944444447) and add it to both sides. Add '0.006944444447' to each side of the equation. 0.1666666667a + 0.006944444447 + a2 = 1 + 0.006944444447 Reorder the terms: 0.006944444447 + 0.1666666667a + a2 = 1 + 0.006944444447 Combine like terms: 1 + 0.006944444447 = 1.006944444447 0.006944444447 + 0.1666666667a + a2 = 1.006944444447 Factor a perfect square on the left side: (a + 0.08333333335)(a + 0.08333333335) = 1.006944444447 Calculate the square root of the right side: 1.003466215 Break this problem into two subproblems by setting (a + 0.08333333335) equal to 1.003466215 and -1.003466215.Subproblem 1
a + 0.08333333335 = 1.003466215 Simplifying a + 0.08333333335 = 1.003466215 Reorder the terms: 0.08333333335 + a = 1.003466215 Solving 0.08333333335 + a = 1.003466215 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + a = 1.003466215 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + a = 1.003466215 + -0.08333333335 a = 1.003466215 + -0.08333333335 Combine like terms: 1.003466215 + -0.08333333335 = 0.92013288165 a = 0.92013288165 Simplifying a = 0.92013288165Subproblem 2
a + 0.08333333335 = -1.003466215 Simplifying a + 0.08333333335 = -1.003466215 Reorder the terms: 0.08333333335 + a = -1.003466215 Solving 0.08333333335 + a = -1.003466215 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.08333333335' to each side of the equation. 0.08333333335 + -0.08333333335 + a = -1.003466215 + -0.08333333335 Combine like terms: 0.08333333335 + -0.08333333335 = 0.00000000000 0.00000000000 + a = -1.003466215 + -0.08333333335 a = -1.003466215 + -0.08333333335 Combine like terms: -1.003466215 + -0.08333333335 = -1.08679954835 a = -1.08679954835 Simplifying a = -1.08679954835Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.92013288165, -1.08679954835}
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