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Simplifying (2e2 + 5e + 7) + -1(e2 + -3e + -1) = 0 Reorder the terms: (7 + 5e + 2e2) + -1(e2 + -3e + -1) = 0 Remove parenthesis around (7 + 5e + 2e2) 7 + 5e + 2e2 + -1(e2 + -3e + -1) = 0 Reorder the terms: 7 + 5e + 2e2 + -1(-1 + -3e + e2) = 0 7 + 5e + 2e2 + (-1 * -1 + -3e * -1 + e2 * -1) = 0 7 + 5e + 2e2 + (1 + 3e + -1e2) = 0 Reorder the terms: 7 + 1 + 5e + 3e + 2e2 + -1e2 = 0 Combine like terms: 7 + 1 = 8 8 + 5e + 3e + 2e2 + -1e2 = 0 Combine like terms: 5e + 3e = 8e 8 + 8e + 2e2 + -1e2 = 0 Combine like terms: 2e2 + -1e2 = 1e2 8 + 8e + 1e2 = 0 Solving 8 + 8e + 1e2 = 0 Solving for variable 'e'. Begin completing the square. Move the constant term to the right: Add '-8' to each side of the equation. 8 + 8e + -8 + e2 = 0 + -8 Reorder the terms: 8 + -8 + 8e + e2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + 8e + e2 = 0 + -8 8e + e2 = 0 + -8 Combine like terms: 0 + -8 = -8 8e + e2 = -8 The e term is 8e. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8e + 16 + e2 = -8 + 16 Reorder the terms: 16 + 8e + e2 = -8 + 16 Combine like terms: -8 + 16 = 8 16 + 8e + e2 = 8 Factor a perfect square on the left side: (e + 4)(e + 4) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (e + 4) equal to 2.828427125 and -2.828427125.Subproblem 1
e + 4 = 2.828427125 Simplifying e + 4 = 2.828427125 Reorder the terms: 4 + e = 2.828427125 Solving 4 + e = 2.828427125 Solving for variable 'e'. Move all terms containing e to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + e = 2.828427125 + -4 Combine like terms: 4 + -4 = 0 0 + e = 2.828427125 + -4 e = 2.828427125 + -4 Combine like terms: 2.828427125 + -4 = -1.171572875 e = -1.171572875 Simplifying e = -1.171572875Subproblem 2
e + 4 = -2.828427125 Simplifying e + 4 = -2.828427125 Reorder the terms: 4 + e = -2.828427125 Solving 4 + e = -2.828427125 Solving for variable 'e'. Move all terms containing e to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + e = -2.828427125 + -4 Combine like terms: 4 + -4 = 0 0 + e = -2.828427125 + -4 e = -2.828427125 + -4 Combine like terms: -2.828427125 + -4 = -6.828427125 e = -6.828427125 Simplifying e = -6.828427125Solution
The solution to the problem is based on the solutions from the subproblems. e = {-1.171572875, -6.828427125}
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