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Simplifying -24a2 + 60a + 30 = 0 Reorder the terms: 30 + 60a + -24a2 = 0 Solving 30 + 60a + -24a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '6'. 6(5 + 10a + -4a2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(5 + 10a + -4a2)' equal to zero and attempt to solve: Simplifying 5 + 10a + -4a2 = 0 Solving 5 + 10a + -4a2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.25 + -2.5a + a2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + -2.5a + 1.25 + a2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + -2.5a + a2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + -2.5a + a2 = 0 + 1.25 -2.5a + a2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 -2.5a + a2 = 1.25 The a term is -2.5a. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5a + 1.5625 + a2 = 1.25 + 1.5625 Reorder the terms: 1.5625 + -2.5a + a2 = 1.25 + 1.5625 Combine like terms: 1.25 + 1.5625 = 2.8125 1.5625 + -2.5a + a2 = 2.8125 Factor a perfect square on the left side: (a + -1.25)(a + -1.25) = 2.8125 Calculate the square root of the right side: 1.677050983 Break this problem into two subproblems by setting (a + -1.25) equal to 1.677050983 and -1.677050983.Subproblem 1
a + -1.25 = 1.677050983 Simplifying a + -1.25 = 1.677050983 Reorder the terms: -1.25 + a = 1.677050983 Solving -1.25 + a = 1.677050983 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = 1.677050983 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = 1.677050983 + 1.25 a = 1.677050983 + 1.25 Combine like terms: 1.677050983 + 1.25 = 2.927050983 a = 2.927050983 Simplifying a = 2.927050983Subproblem 2
a + -1.25 = -1.677050983 Simplifying a + -1.25 = -1.677050983 Reorder the terms: -1.25 + a = -1.677050983 Solving -1.25 + a = -1.677050983 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + a = -1.677050983 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + a = -1.677050983 + 1.25 a = -1.677050983 + 1.25 Combine like terms: -1.677050983 + 1.25 = -0.427050983 a = -0.427050983 Simplifying a = -0.427050983Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.927050983, -0.427050983}Solution
a = {2.927050983, -0.427050983}
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