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Simplifying 3a + a2 = 20 Solving 3a + a2 = 20 Solving for variable 'a'. Reorder the terms: -20 + 3a + a2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + 3a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '20' to each side of the equation. -20 + 3a + 20 + a2 = 0 + 20 Reorder the terms: -20 + 20 + 3a + a2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 3a + a2 = 0 + 20 3a + a2 = 0 + 20 Combine like terms: 0 + 20 = 20 3a + a2 = 20 The a term is 3a. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3a + 2.25 + a2 = 20 + 2.25 Reorder the terms: 2.25 + 3a + a2 = 20 + 2.25 Combine like terms: 20 + 2.25 = 22.25 2.25 + 3a + a2 = 22.25 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 22.25 Calculate the square root of the right side: 4.716990566 Break this problem into two subproblems by setting (a + 1.5) equal to 4.716990566 and -4.716990566.Subproblem 1
a + 1.5 = 4.716990566 Simplifying a + 1.5 = 4.716990566 Reorder the terms: 1.5 + a = 4.716990566 Solving 1.5 + a = 4.716990566 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 4.716990566 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 4.716990566 + -1.5 a = 4.716990566 + -1.5 Combine like terms: 4.716990566 + -1.5 = 3.216990566 a = 3.216990566 Simplifying a = 3.216990566Subproblem 2
a + 1.5 = -4.716990566 Simplifying a + 1.5 = -4.716990566 Reorder the terms: 1.5 + a = -4.716990566 Solving 1.5 + a = -4.716990566 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -4.716990566 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -4.716990566 + -1.5 a = -4.716990566 + -1.5 Combine like terms: -4.716990566 + -1.5 = -6.216990566 a = -6.216990566 Simplifying a = -6.216990566Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.216990566, -6.216990566}
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