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Simplifying 2a2 + 3a + -7 = 30 Reorder the terms: -7 + 3a + 2a2 = 30 Solving -7 + 3a + 2a2 = 30 Solving for variable 'a'. Reorder the terms: -7 + -30 + 3a + 2a2 = 30 + -30 Combine like terms: -7 + -30 = -37 -37 + 3a + 2a2 = 30 + -30 Combine like terms: 30 + -30 = 0 -37 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -18.5 + 1.5a + a2 = 0 Move the constant term to the right: Add '18.5' to each side of the equation. -18.5 + 1.5a + 18.5 + a2 = 0 + 18.5 Reorder the terms: -18.5 + 18.5 + 1.5a + a2 = 0 + 18.5 Combine like terms: -18.5 + 18.5 = 0.0 0.0 + 1.5a + a2 = 0 + 18.5 1.5a + a2 = 0 + 18.5 Combine like terms: 0 + 18.5 = 18.5 1.5a + a2 = 18.5 The a term is 1.5a. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5a + 0.5625 + a2 = 18.5 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 18.5 + 0.5625 Combine like terms: 18.5 + 0.5625 = 19.0625 0.5625 + 1.5a + a2 = 19.0625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 19.0625 Calculate the square root of the right side: 4.366062299 Break this problem into two subproblems by setting (a + 0.75) equal to 4.366062299 and -4.366062299.Subproblem 1
a + 0.75 = 4.366062299 Simplifying a + 0.75 = 4.366062299 Reorder the terms: 0.75 + a = 4.366062299 Solving 0.75 + a = 4.366062299 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 4.366062299 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 4.366062299 + -0.75 a = 4.366062299 + -0.75 Combine like terms: 4.366062299 + -0.75 = 3.616062299 a = 3.616062299 Simplifying a = 3.616062299Subproblem 2
a + 0.75 = -4.366062299 Simplifying a + 0.75 = -4.366062299 Reorder the terms: 0.75 + a = -4.366062299 Solving 0.75 + a = -4.366062299 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -4.366062299 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -4.366062299 + -0.75 a = -4.366062299 + -0.75 Combine like terms: -4.366062299 + -0.75 = -5.116062299 a = -5.116062299 Simplifying a = -5.116062299Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.616062299, -5.116062299}
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