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Simplifying 40a2 + 20a + -30 = 0 Reorder the terms: -30 + 20a + 40a2 = 0 Solving -30 + 20a + 40a2 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '10'. 10(-3 + 2a + 4a2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(-3 + 2a + 4a2)' equal to zero and attempt to solve: Simplifying -3 + 2a + 4a2 = 0 Solving -3 + 2a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 0.5a + a2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.5a + 0.75 + a2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.5a + a2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.5a + a2 = 0 + 0.75 0.5a + a2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.5a + a2 = 0.75 The a term is 0.5a. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5a + 0.0625 + a2 = 0.75 + 0.0625 Reorder the terms: 0.0625 + 0.5a + a2 = 0.75 + 0.0625 Combine like terms: 0.75 + 0.0625 = 0.8125 0.0625 + 0.5a + a2 = 0.8125 Factor a perfect square on the left side: (a + 0.25)(a + 0.25) = 0.8125 Calculate the square root of the right side: 0.901387819 Break this problem into two subproblems by setting (a + 0.25) equal to 0.901387819 and -0.901387819.Subproblem 1
a + 0.25 = 0.901387819 Simplifying a + 0.25 = 0.901387819 Reorder the terms: 0.25 + a = 0.901387819 Solving 0.25 + a = 0.901387819 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = 0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = 0.901387819 + -0.25 a = 0.901387819 + -0.25 Combine like terms: 0.901387819 + -0.25 = 0.651387819 a = 0.651387819 Simplifying a = 0.651387819Subproblem 2
a + 0.25 = -0.901387819 Simplifying a + 0.25 = -0.901387819 Reorder the terms: 0.25 + a = -0.901387819 Solving 0.25 + a = -0.901387819 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = -0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = -0.901387819 + -0.25 a = -0.901387819 + -0.25 Combine like terms: -0.901387819 + -0.25 = -1.151387819 a = -1.151387819 Simplifying a = -1.151387819Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.651387819, -1.151387819}Solution
a = {0.651387819, -1.151387819}
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