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Simplifying 40a2 + -32a = 30 Reorder the terms: -32a + 40a2 = 30 Solving -32a + 40a2 = 30 Solving for variable 'a'. Reorder the terms: -30 + -32a + 40a2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + -32a + 40a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-15 + -16a + 20a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-15 + -16a + 20a2)' equal to zero and attempt to solve: Simplifying -15 + -16a + 20a2 = 0 Solving -15 + -16a + 20a2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -0.75 + -0.8a + a2 = 0.0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.8a + 0.75 + a2 = 0.0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.8a + a2 = 0.0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.8a + a2 = 0.0 + 0.75 -0.8a + a2 = 0.0 + 0.75 Combine like terms: 0.0 + 0.75 = 0.75 -0.8a + a2 = 0.75 The a term is -0.8a. Take half its coefficient (-0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. -0.8a + 0.16 + a2 = 0.75 + 0.16 Reorder the terms: 0.16 + -0.8a + a2 = 0.75 + 0.16 Combine like terms: 0.75 + 0.16 = 0.91 0.16 + -0.8a + a2 = 0.91 Factor a perfect square on the left side: (a + -0.4)(a + -0.4) = 0.91 Calculate the square root of the right side: 0.953939201 Break this problem into two subproblems by setting (a + -0.4) equal to 0.953939201 and -0.953939201.Subproblem 1
a + -0.4 = 0.953939201 Simplifying a + -0.4 = 0.953939201 Reorder the terms: -0.4 + a = 0.953939201 Solving -0.4 + a = 0.953939201 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + a = 0.953939201 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + a = 0.953939201 + 0.4 a = 0.953939201 + 0.4 Combine like terms: 0.953939201 + 0.4 = 1.353939201 a = 1.353939201 Simplifying a = 1.353939201Subproblem 2
a + -0.4 = -0.953939201 Simplifying a + -0.4 = -0.953939201 Reorder the terms: -0.4 + a = -0.953939201 Solving -0.4 + a = -0.953939201 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.4' to each side of the equation. -0.4 + 0.4 + a = -0.953939201 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + a = -0.953939201 + 0.4 a = -0.953939201 + 0.4 Combine like terms: -0.953939201 + 0.4 = -0.553939201 a = -0.553939201 Simplifying a = -0.553939201Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.353939201, -0.553939201}Solution
a = {1.353939201, -0.553939201}
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