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Simplifying 40a2 + -42a + -9 = 0 Reorder the terms: -9 + -42a + 40a2 = 0 Solving -9 + -42a + 40a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -0.225 + -1.05a + a2 = 0 Move the constant term to the right: Add '0.225' to each side of the equation. -0.225 + -1.05a + 0.225 + a2 = 0 + 0.225 Reorder the terms: -0.225 + 0.225 + -1.05a + a2 = 0 + 0.225 Combine like terms: -0.225 + 0.225 = 0.000 0.000 + -1.05a + a2 = 0 + 0.225 -1.05a + a2 = 0 + 0.225 Combine like terms: 0 + 0.225 = 0.225 -1.05a + a2 = 0.225 The a term is -1.05a. Take half its coefficient (-0.525). Square it (0.275625) and add it to both sides. Add '0.275625' to each side of the equation. -1.05a + 0.275625 + a2 = 0.225 + 0.275625 Reorder the terms: 0.275625 + -1.05a + a2 = 0.225 + 0.275625 Combine like terms: 0.225 + 0.275625 = 0.500625 0.275625 + -1.05a + a2 = 0.500625 Factor a perfect square on the left side: (a + -0.525)(a + -0.525) = 0.500625 Calculate the square root of the right side: 0.707548585 Break this problem into two subproblems by setting (a + -0.525) equal to 0.707548585 and -0.707548585.Subproblem 1
a + -0.525 = 0.707548585 Simplifying a + -0.525 = 0.707548585 Reorder the terms: -0.525 + a = 0.707548585 Solving -0.525 + a = 0.707548585 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.525' to each side of the equation. -0.525 + 0.525 + a = 0.707548585 + 0.525 Combine like terms: -0.525 + 0.525 = 0.000 0.000 + a = 0.707548585 + 0.525 a = 0.707548585 + 0.525 Combine like terms: 0.707548585 + 0.525 = 1.232548585 a = 1.232548585 Simplifying a = 1.232548585Subproblem 2
a + -0.525 = -0.707548585 Simplifying a + -0.525 = -0.707548585 Reorder the terms: -0.525 + a = -0.707548585 Solving -0.525 + a = -0.707548585 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.525' to each side of the equation. -0.525 + 0.525 + a = -0.707548585 + 0.525 Combine like terms: -0.525 + 0.525 = 0.000 0.000 + a = -0.707548585 + 0.525 a = -0.707548585 + 0.525 Combine like terms: -0.707548585 + 0.525 = -0.182548585 a = -0.182548585 Simplifying a = -0.182548585Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.232548585, -0.182548585}
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