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Simplifying 7a2 = 7 + -40a Solving 7a2 = 7 + -40a Solving for variable 'a'. Reorder the terms: -7 + 40a + 7a2 = 7 + -40a + -7 + 40a Reorder the terms: -7 + 40a + 7a2 = 7 + -7 + -40a + 40a Combine like terms: 7 + -7 = 0 -7 + 40a + 7a2 = 0 + -40a + 40a -7 + 40a + 7a2 = -40a + 40a Combine like terms: -40a + 40a = 0 -7 + 40a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1 + 5.714285714a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 5.714285714a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + 5.714285714a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 5.714285714a + a2 = 0 + 1 5.714285714a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 5.714285714a + a2 = 1 The a term is 5.714285714a. Take half its coefficient (2.857142857). Square it (8.163265305) and add it to both sides. Add '8.163265305' to each side of the equation. 5.714285714a + 8.163265305 + a2 = 1 + 8.163265305 Reorder the terms: 8.163265305 + 5.714285714a + a2 = 1 + 8.163265305 Combine like terms: 1 + 8.163265305 = 9.163265305 8.163265305 + 5.714285714a + a2 = 9.163265305 Factor a perfect square on the left side: (a + 2.857142857)(a + 2.857142857) = 9.163265305 Calculate the square root of the right side: 3.027088586 Break this problem into two subproblems by setting (a + 2.857142857) equal to 3.027088586 and -3.027088586.Subproblem 1
a + 2.857142857 = 3.027088586 Simplifying a + 2.857142857 = 3.027088586 Reorder the terms: 2.857142857 + a = 3.027088586 Solving 2.857142857 + a = 3.027088586 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.857142857' to each side of the equation. 2.857142857 + -2.857142857 + a = 3.027088586 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + a = 3.027088586 + -2.857142857 a = 3.027088586 + -2.857142857 Combine like terms: 3.027088586 + -2.857142857 = 0.169945729 a = 0.169945729 Simplifying a = 0.169945729Subproblem 2
a + 2.857142857 = -3.027088586 Simplifying a + 2.857142857 = -3.027088586 Reorder the terms: 2.857142857 + a = -3.027088586 Solving 2.857142857 + a = -3.027088586 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.857142857' to each side of the equation. 2.857142857 + -2.857142857 + a = -3.027088586 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + a = -3.027088586 + -2.857142857 a = -3.027088586 + -2.857142857 Combine like terms: -3.027088586 + -2.857142857 = -5.884231443 a = -5.884231443 Simplifying a = -5.884231443Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.169945729, -5.884231443}
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