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Simplifying (5a2 + -3a + 9) + -1(20a2 + 11a + -6) = 0 Reorder the terms: (9 + -3a + 5a2) + -1(20a2 + 11a + -6) = 0 Remove parenthesis around (9 + -3a + 5a2) 9 + -3a + 5a2 + -1(20a2 + 11a + -6) = 0 Reorder the terms: 9 + -3a + 5a2 + -1(-6 + 11a + 20a2) = 0 9 + -3a + 5a2 + (-6 * -1 + 11a * -1 + 20a2 * -1) = 0 9 + -3a + 5a2 + (6 + -11a + -20a2) = 0 Reorder the terms: 9 + 6 + -3a + -11a + 5a2 + -20a2 = 0 Combine like terms: 9 + 6 = 15 15 + -3a + -11a + 5a2 + -20a2 = 0 Combine like terms: -3a + -11a = -14a 15 + -14a + 5a2 + -20a2 = 0 Combine like terms: 5a2 + -20a2 = -15a2 15 + -14a + -15a2 = 0 Solving 15 + -14a + -15a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. -1 + 0.9333333333a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.9333333333a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + 0.9333333333a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.9333333333a + a2 = 0 + 1 0.9333333333a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.9333333333a + a2 = 1 The a term is 0.9333333333a. Take half its coefficient (0.4666666667). Square it (0.2177777778) and add it to both sides. Add '0.2177777778' to each side of the equation. 0.9333333333a + 0.2177777778 + a2 = 1 + 0.2177777778 Reorder the terms: 0.2177777778 + 0.9333333333a + a2 = 1 + 0.2177777778 Combine like terms: 1 + 0.2177777778 = 1.2177777778 0.2177777778 + 0.9333333333a + a2 = 1.2177777778 Factor a perfect square on the left side: (a + 0.4666666667)(a + 0.4666666667) = 1.2177777778 Calculate the square root of the right side: 1.10352969 Break this problem into two subproblems by setting (a + 0.4666666667) equal to 1.10352969 and -1.10352969.Subproblem 1
a + 0.4666666667 = 1.10352969 Simplifying a + 0.4666666667 = 1.10352969 Reorder the terms: 0.4666666667 + a = 1.10352969 Solving 0.4666666667 + a = 1.10352969 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4666666667' to each side of the equation. 0.4666666667 + -0.4666666667 + a = 1.10352969 + -0.4666666667 Combine like terms: 0.4666666667 + -0.4666666667 = 0.0000000000 0.0000000000 + a = 1.10352969 + -0.4666666667 a = 1.10352969 + -0.4666666667 Combine like terms: 1.10352969 + -0.4666666667 = 0.6368630233 a = 0.6368630233 Simplifying a = 0.6368630233Subproblem 2
a + 0.4666666667 = -1.10352969 Simplifying a + 0.4666666667 = -1.10352969 Reorder the terms: 0.4666666667 + a = -1.10352969 Solving 0.4666666667 + a = -1.10352969 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4666666667' to each side of the equation. 0.4666666667 + -0.4666666667 + a = -1.10352969 + -0.4666666667 Combine like terms: 0.4666666667 + -0.4666666667 = 0.0000000000 0.0000000000 + a = -1.10352969 + -0.4666666667 a = -1.10352969 + -0.4666666667 Combine like terms: -1.10352969 + -0.4666666667 = -1.5701963567 a = -1.5701963567 Simplifying a = -1.5701963567Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.6368630233, -1.5701963567}
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