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Simplifying a(5 + 3a) = 22 + -7 (5 * a + 3a * a) = 22 + -7 (5a + 3a2) = 22 + -7 Combine like terms: 22 + -7 = 15 5a + 3a2 = 15 Solving 5a + 3a2 = 15 Solving for variable 'a'. Reorder the terms: -15 + 5a + 3a2 = 15 + -15 Combine like terms: 15 + -15 = 0 -15 + 5a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -5 + 1.666666667a + a2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + 1.666666667a + 5 + a2 = 0 + 5 Reorder the terms: -5 + 5 + 1.666666667a + a2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 1.666666667a + a2 = 0 + 5 1.666666667a + a2 = 0 + 5 Combine like terms: 0 + 5 = 5 1.666666667a + a2 = 5 The a term is 1.666666667a. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667a + 0.6944444447 + a2 = 5 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667a + a2 = 5 + 0.6944444447 Combine like terms: 5 + 0.6944444447 = 5.6944444447 0.6944444447 + 1.666666667a + a2 = 5.6944444447 Factor a perfect square on the left side: (a + 0.8333333335)(a + 0.8333333335) = 5.6944444447 Calculate the square root of the right side: 2.386303511 Break this problem into two subproblems by setting (a + 0.8333333335) equal to 2.386303511 and -2.386303511.Subproblem 1
a + 0.8333333335 = 2.386303511 Simplifying a + 0.8333333335 = 2.386303511 Reorder the terms: 0.8333333335 + a = 2.386303511 Solving 0.8333333335 + a = 2.386303511 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + a = 2.386303511 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = 2.386303511 + -0.8333333335 a = 2.386303511 + -0.8333333335 Combine like terms: 2.386303511 + -0.8333333335 = 1.5529701775 a = 1.5529701775 Simplifying a = 1.5529701775Subproblem 2
a + 0.8333333335 = -2.386303511 Simplifying a + 0.8333333335 = -2.386303511 Reorder the terms: 0.8333333335 + a = -2.386303511 Solving 0.8333333335 + a = -2.386303511 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + a = -2.386303511 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = -2.386303511 + -0.8333333335 a = -2.386303511 + -0.8333333335 Combine like terms: -2.386303511 + -0.8333333335 = -3.2196368445 a = -3.2196368445 Simplifying a = -3.2196368445Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.5529701775, -3.2196368445}
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