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Simplifying (5a + -3a2) + (8 + 7a) = 0 Remove parenthesis around (5a + -3a2) 5a + -3a2 + (8 + 7a) = 0 Remove parenthesis around (8 + 7a) 5a + -3a2 + 8 + 7a = 0 Reorder the terms: 8 + 5a + 7a + -3a2 = 0 Combine like terms: 5a + 7a = 12a 8 + 12a + -3a2 = 0 Solving 8 + 12a + -3a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -2.666666667 + -4a + a2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + -4a + 2.666666667 + a2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + -4a + a2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + -4a + a2 = 0 + 2.666666667 -4a + a2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 -4a + a2 = 2.666666667 The a term is -4a. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4a + 4 + a2 = 2.666666667 + 4 Reorder the terms: 4 + -4a + a2 = 2.666666667 + 4 Combine like terms: 2.666666667 + 4 = 6.666666667 4 + -4a + a2 = 6.666666667 Factor a perfect square on the left side: (a + -2)(a + -2) = 6.666666667 Calculate the square root of the right side: 2.581988898 Break this problem into two subproblems by setting (a + -2) equal to 2.581988898 and -2.581988898.Subproblem 1
a + -2 = 2.581988898 Simplifying a + -2 = 2.581988898 Reorder the terms: -2 + a = 2.581988898 Solving -2 + a = 2.581988898 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = 2.581988898 + 2 Combine like terms: -2 + 2 = 0 0 + a = 2.581988898 + 2 a = 2.581988898 + 2 Combine like terms: 2.581988898 + 2 = 4.581988898 a = 4.581988898 Simplifying a = 4.581988898Subproblem 2
a + -2 = -2.581988898 Simplifying a + -2 = -2.581988898 Reorder the terms: -2 + a = -2.581988898 Solving -2 + a = -2.581988898 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = -2.581988898 + 2 Combine like terms: -2 + 2 = 0 0 + a = -2.581988898 + 2 a = -2.581988898 + 2 Combine like terms: -2.581988898 + 2 = -0.581988898 a = -0.581988898 Simplifying a = -0.581988898Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.581988898, -0.581988898}
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