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Simplifying 7a2 + 32 = -40a Reorder the terms: 32 + 7a2 = -40a Solving 32 + 7a2 = -40a Solving for variable 'a'. Reorder the terms: 32 + 40a + 7a2 = -40a + 40a Combine like terms: -40a + 40a = 0 32 + 40a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 4.571428571 + 5.714285714a + a2 = 0 Move the constant term to the right: Add '-4.571428571' to each side of the equation. 4.571428571 + 5.714285714a + -4.571428571 + a2 = 0 + -4.571428571 Reorder the terms: 4.571428571 + -4.571428571 + 5.714285714a + a2 = 0 + -4.571428571 Combine like terms: 4.571428571 + -4.571428571 = 0.000000000 0.000000000 + 5.714285714a + a2 = 0 + -4.571428571 5.714285714a + a2 = 0 + -4.571428571 Combine like terms: 0 + -4.571428571 = -4.571428571 5.714285714a + a2 = -4.571428571 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 = -4.571428571 + 8.163265305 Reorder the terms: 8.163265305 + 5.714285714a + a2 = -4.571428571 + 8.163265305 Combine like terms: -4.571428571 + 8.163265305 = 3.591836734 8.163265305 + 5.714285714a + a2 = 3.591836734 Factor a perfect square on the left side: (a + 2.857142857)(a + 2.857142857) = 3.591836734 Calculate the square root of the right side: 1.895214166 Break this problem into two subproblems by setting (a + 2.857142857) equal to 1.895214166 and -1.895214166.Subproblem 1
a + 2.857142857 = 1.895214166 Simplifying a + 2.857142857 = 1.895214166 Reorder the terms: 2.857142857 + a = 1.895214166 Solving 2.857142857 + a = 1.895214166 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 = 1.895214166 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + a = 1.895214166 + -2.857142857 a = 1.895214166 + -2.857142857 Combine like terms: 1.895214166 + -2.857142857 = -0.961928691 a = -0.961928691 Simplifying a = -0.961928691Subproblem 2
a + 2.857142857 = -1.895214166 Simplifying a + 2.857142857 = -1.895214166 Reorder the terms: 2.857142857 + a = -1.895214166 Solving 2.857142857 + a = -1.895214166 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 = -1.895214166 + -2.857142857 Combine like terms: 2.857142857 + -2.857142857 = 0.000000000 0.000000000 + a = -1.895214166 + -2.857142857 a = -1.895214166 + -2.857142857 Combine like terms: -1.895214166 + -2.857142857 = -4.752357023 a = -4.752357023 Simplifying a = -4.752357023Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.961928691, -4.752357023}
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