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Simplifying 7a2 + 32a = 7 + -40a Reorder the terms: 32a + 7a2 = 7 + -40a Solving 32a + 7a2 = 7 + -40a Solving for variable 'a'. Reorder the terms: -7 + 32a + 40a + 7a2 = 7 + -40a + -7 + 40a Combine like terms: 32a + 40a = 72a -7 + 72a + 7a2 = 7 + -40a + -7 + 40a Reorder the terms: -7 + 72a + 7a2 = 7 + -7 + -40a + 40a Combine like terms: 7 + -7 = 0 -7 + 72a + 7a2 = 0 + -40a + 40a -7 + 72a + 7a2 = -40a + 40a Combine like terms: -40a + 40a = 0 -7 + 72a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1 + 10.28571429a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 10.28571429a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + 10.28571429a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 10.28571429a + a2 = 0 + 1 10.28571429a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 10.28571429a + a2 = 1 The a term is 10.28571429a. Take half its coefficient (5.142857145). Square it (26.44897961) and add it to both sides. Add '26.44897961' to each side of the equation. 10.28571429a + 26.44897961 + a2 = 1 + 26.44897961 Reorder the terms: 26.44897961 + 10.28571429a + a2 = 1 + 26.44897961 Combine like terms: 1 + 26.44897961 = 27.44897961 26.44897961 + 10.28571429a + a2 = 27.44897961 Factor a perfect square on the left side: (a + 5.142857145)(a + 5.142857145) = 27.44897961 Calculate the square root of the right side: 5.239177379 Break this problem into two subproblems by setting (a + 5.142857145) equal to 5.239177379 and -5.239177379.Subproblem 1
a + 5.142857145 = 5.239177379 Simplifying a + 5.142857145 = 5.239177379 Reorder the terms: 5.142857145 + a = 5.239177379 Solving 5.142857145 + a = 5.239177379 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-5.142857145' to each side of the equation. 5.142857145 + -5.142857145 + a = 5.239177379 + -5.142857145 Combine like terms: 5.142857145 + -5.142857145 = 0.000000000 0.000000000 + a = 5.239177379 + -5.142857145 a = 5.239177379 + -5.142857145 Combine like terms: 5.239177379 + -5.142857145 = 0.096320234 a = 0.096320234 Simplifying a = 0.096320234Subproblem 2
a + 5.142857145 = -5.239177379 Simplifying a + 5.142857145 = -5.239177379 Reorder the terms: 5.142857145 + a = -5.239177379 Solving 5.142857145 + a = -5.239177379 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-5.142857145' to each side of the equation. 5.142857145 + -5.142857145 + a = -5.239177379 + -5.142857145 Combine like terms: 5.142857145 + -5.142857145 = 0.000000000 0.000000000 + a = -5.239177379 + -5.142857145 a = -5.239177379 + -5.142857145 Combine like terms: -5.239177379 + -5.142857145 = -10.382034524 a = -10.382034524 Simplifying a = -10.382034524Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.096320234, -10.382034524}
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