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Simplifying -15a2 + 30a + 462 = 0 Reorder the terms: 462 + 30a + -15a2 = 0 Solving 462 + 30a + -15a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '3'. 3(154 + 10a + -5a2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(154 + 10a + -5a2)' equal to zero and attempt to solve: Simplifying 154 + 10a + -5a2 = 0 Solving 154 + 10a + -5a2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -30.8 + -2a + a2 = 0 Move the constant term to the right: Add '30.8' to each side of the equation. -30.8 + -2a + 30.8 + a2 = 0 + 30.8 Reorder the terms: -30.8 + 30.8 + -2a + a2 = 0 + 30.8 Combine like terms: -30.8 + 30.8 = 0.0 0.0 + -2a + a2 = 0 + 30.8 -2a + a2 = 0 + 30.8 Combine like terms: 0 + 30.8 = 30.8 -2a + a2 = 30.8 The a term is -2a. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2a + 1 + a2 = 30.8 + 1 Reorder the terms: 1 + -2a + a2 = 30.8 + 1 Combine like terms: 30.8 + 1 = 31.8 1 + -2a + a2 = 31.8 Factor a perfect square on the left side: (a + -1)(a + -1) = 31.8 Calculate the square root of the right side: 5.639148872 Break this problem into two subproblems by setting (a + -1) equal to 5.639148872 and -5.639148872.Subproblem 1
a + -1 = 5.639148872 Simplifying a + -1 = 5.639148872 Reorder the terms: -1 + a = 5.639148872 Solving -1 + a = 5.639148872 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + a = 5.639148872 + 1 Combine like terms: -1 + 1 = 0 0 + a = 5.639148872 + 1 a = 5.639148872 + 1 Combine like terms: 5.639148872 + 1 = 6.639148872 a = 6.639148872 Simplifying a = 6.639148872Subproblem 2
a + -1 = -5.639148872 Simplifying a + -1 = -5.639148872 Reorder the terms: -1 + a = -5.639148872 Solving -1 + a = -5.639148872 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + a = -5.639148872 + 1 Combine like terms: -1 + 1 = 0 0 + a = -5.639148872 + 1 a = -5.639148872 + 1 Combine like terms: -5.639148872 + 1 = -4.639148872 a = -4.639148872 Simplifying a = -4.639148872Solution
The solution to the problem is based on the solutions from the subproblems. a = {6.639148872, -4.639148872}Solution
a = {6.639148872, -4.639148872}
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