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Simplifying -4a2 + 18a + 120 = 0 Reorder the terms: 120 + 18a + -4a2 = 0 Solving 120 + 18a + -4a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2'. 2(60 + 9a + -2a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(60 + 9a + -2a2)' equal to zero and attempt to solve: Simplifying 60 + 9a + -2a2 = 0 Solving 60 + 9a + -2a2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -30 + -4.5a + a2 = 0 Move the constant term to the right: Add '30' to each side of the equation. -30 + -4.5a + 30 + a2 = 0 + 30 Reorder the terms: -30 + 30 + -4.5a + a2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + -4.5a + a2 = 0 + 30 -4.5a + a2 = 0 + 30 Combine like terms: 0 + 30 = 30 -4.5a + a2 = 30 The a term is -4.5a. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5a + 5.0625 + a2 = 30 + 5.0625 Reorder the terms: 5.0625 + -4.5a + a2 = 30 + 5.0625 Combine like terms: 30 + 5.0625 = 35.0625 5.0625 + -4.5a + a2 = 35.0625 Factor a perfect square on the left side: (a + -2.25)(a + -2.25) = 35.0625 Calculate the square root of the right side: 5.921359641 Break this problem into two subproblems by setting (a + -2.25) equal to 5.921359641 and -5.921359641.Subproblem 1
a + -2.25 = 5.921359641 Simplifying a + -2.25 = 5.921359641 Reorder the terms: -2.25 + a = 5.921359641 Solving -2.25 + a = 5.921359641 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + a = 5.921359641 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = 5.921359641 + 2.25 a = 5.921359641 + 2.25 Combine like terms: 5.921359641 + 2.25 = 8.171359641 a = 8.171359641 Simplifying a = 8.171359641Subproblem 2
a + -2.25 = -5.921359641 Simplifying a + -2.25 = -5.921359641 Reorder the terms: -2.25 + a = -5.921359641 Solving -2.25 + a = -5.921359641 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + a = -5.921359641 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + a = -5.921359641 + 2.25 a = -5.921359641 + 2.25 Combine like terms: -5.921359641 + 2.25 = -3.671359641 a = -3.671359641 Simplifying a = -3.671359641Solution
The solution to the problem is based on the solutions from the subproblems. a = {8.171359641, -3.671359641}Solution
a = {8.171359641, -3.671359641}
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