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Simplifying 2a2 + 3a + -7 = 35 Reorder the terms: -7 + 3a + 2a2 = 35 Solving -7 + 3a + 2a2 = 35 Solving for variable 'a'. Reorder the terms: -7 + -35 + 3a + 2a2 = 35 + -35 Combine like terms: -7 + -35 = -42 -42 + 3a + 2a2 = 35 + -35 Combine like terms: 35 + -35 = 0 -42 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -21 + 1.5a + a2 = 0 Move the constant term to the right: Add '21' to each side of the equation. -21 + 1.5a + 21 + a2 = 0 + 21 Reorder the terms: -21 + 21 + 1.5a + a2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 1.5a + a2 = 0 + 21 1.5a + a2 = 0 + 21 Combine like terms: 0 + 21 = 21 1.5a + a2 = 21 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 = 21 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 21 + 0.5625 Combine like terms: 21 + 0.5625 = 21.5625 0.5625 + 1.5a + a2 = 21.5625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 21.5625 Calculate the square root of the right side: 4.643543905 Break this problem into two subproblems by setting (a + 0.75) equal to 4.643543905 and -4.643543905.Subproblem 1
a + 0.75 = 4.643543905 Simplifying a + 0.75 = 4.643543905 Reorder the terms: 0.75 + a = 4.643543905 Solving 0.75 + a = 4.643543905 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 = 4.643543905 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 4.643543905 + -0.75 a = 4.643543905 + -0.75 Combine like terms: 4.643543905 + -0.75 = 3.893543905 a = 3.893543905 Simplifying a = 3.893543905Subproblem 2
a + 0.75 = -4.643543905 Simplifying a + 0.75 = -4.643543905 Reorder the terms: 0.75 + a = -4.643543905 Solving 0.75 + a = -4.643543905 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 = -4.643543905 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -4.643543905 + -0.75 a = -4.643543905 + -0.75 Combine like terms: -4.643543905 + -0.75 = -5.393543905 a = -5.393543905 Simplifying a = -5.393543905Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.893543905, -5.393543905}
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