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Simplifying 3a(5 + 9a) + (4a + -12) = 0 (5 * 3a + 9a * 3a) + (4a + -12) = 0 (15a + 27a2) + (4a + -12) = 0 Reorder the terms: 15a + 27a2 + (-12 + 4a) = 0 Remove parenthesis around (-12 + 4a) 15a + 27a2 + -12 + 4a = 0 Reorder the terms: -12 + 15a + 4a + 27a2 = 0 Combine like terms: 15a + 4a = 19a -12 + 19a + 27a2 = 0 Solving -12 + 19a + 27a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -0.4444444444 + 0.7037037037a + a2 = 0 Move the constant term to the right: Add '0.4444444444' to each side of the equation. -0.4444444444 + 0.7037037037a + 0.4444444444 + a2 = 0 + 0.4444444444 Reorder the terms: -0.4444444444 + 0.4444444444 + 0.7037037037a + a2 = 0 + 0.4444444444 Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000 0.0000000000 + 0.7037037037a + a2 = 0 + 0.4444444444 0.7037037037a + a2 = 0 + 0.4444444444 Combine like terms: 0 + 0.4444444444 = 0.4444444444 0.7037037037a + a2 = 0.4444444444 The a term is 0.7037037037a. Take half its coefficient (0.3518518519). Square it (0.1237997257) and add it to both sides. Add '0.1237997257' to each side of the equation. 0.7037037037a + 0.1237997257 + a2 = 0.4444444444 + 0.1237997257 Reorder the terms: 0.1237997257 + 0.7037037037a + a2 = 0.4444444444 + 0.1237997257 Combine like terms: 0.4444444444 + 0.1237997257 = 0.5682441701 0.1237997257 + 0.7037037037a + a2 = 0.5682441701 Factor a perfect square on the left side: (a + 0.3518518519)(a + 0.3518518519) = 0.5682441701 Calculate the square root of the right side: 0.75381972 Break this problem into two subproblems by setting (a + 0.3518518519) equal to 0.75381972 and -0.75381972.Subproblem 1
a + 0.3518518519 = 0.75381972 Simplifying a + 0.3518518519 = 0.75381972 Reorder the terms: 0.3518518519 + a = 0.75381972 Solving 0.3518518519 + a = 0.75381972 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3518518519' to each side of the equation. 0.3518518519 + -0.3518518519 + a = 0.75381972 + -0.3518518519 Combine like terms: 0.3518518519 + -0.3518518519 = 0.0000000000 0.0000000000 + a = 0.75381972 + -0.3518518519 a = 0.75381972 + -0.3518518519 Combine like terms: 0.75381972 + -0.3518518519 = 0.4019678681 a = 0.4019678681 Simplifying a = 0.4019678681Subproblem 2
a + 0.3518518519 = -0.75381972 Simplifying a + 0.3518518519 = -0.75381972 Reorder the terms: 0.3518518519 + a = -0.75381972 Solving 0.3518518519 + a = -0.75381972 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.3518518519' to each side of the equation. 0.3518518519 + -0.3518518519 + a = -0.75381972 + -0.3518518519 Combine like terms: 0.3518518519 + -0.3518518519 = 0.0000000000 0.0000000000 + a = -0.75381972 + -0.3518518519 a = -0.75381972 + -0.3518518519 Combine like terms: -0.75381972 + -0.3518518519 = -1.1056715719 a = -1.1056715719 Simplifying a = -1.1056715719Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.4019678681, -1.1056715719}
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