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Simplifying 10e2 + -2e + 7 + -6e2 + 15e = 0 Reorder the terms: 7 + -2e + 15e + 10e2 + -6e2 = 0 Combine like terms: -2e + 15e = 13e 7 + 13e + 10e2 + -6e2 = 0 Combine like terms: 10e2 + -6e2 = 4e2 7 + 13e + 4e2 = 0 Solving 7 + 13e + 4e2 = 0 Solving for variable 'e'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.75 + 3.25e + e2 = 0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + 3.25e + -1.75 + e2 = 0 + -1.75 Reorder the terms: 1.75 + -1.75 + 3.25e + e2 = 0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + 3.25e + e2 = 0 + -1.75 3.25e + e2 = 0 + -1.75 Combine like terms: 0 + -1.75 = -1.75 3.25e + e2 = -1.75 The e term is 3.25e. Take half its coefficient (1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. 3.25e + 2.640625 + e2 = -1.75 + 2.640625 Reorder the terms: 2.640625 + 3.25e + e2 = -1.75 + 2.640625 Combine like terms: -1.75 + 2.640625 = 0.890625 2.640625 + 3.25e + e2 = 0.890625 Factor a perfect square on the left side: (e + 1.625)(e + 1.625) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (e + 1.625) equal to 0.943729304 and -0.943729304.Subproblem 1
e + 1.625 = 0.943729304 Simplifying e + 1.625 = 0.943729304 Reorder the terms: 1.625 + e = 0.943729304 Solving 1.625 + e = 0.943729304 Solving for variable 'e'. Move all terms containing e to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + e = 0.943729304 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + e = 0.943729304 + -1.625 e = 0.943729304 + -1.625 Combine like terms: 0.943729304 + -1.625 = -0.681270696 e = -0.681270696 Simplifying e = -0.681270696Subproblem 2
e + 1.625 = -0.943729304 Simplifying e + 1.625 = -0.943729304 Reorder the terms: 1.625 + e = -0.943729304 Solving 1.625 + e = -0.943729304 Solving for variable 'e'. Move all terms containing e to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + e = -0.943729304 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + e = -0.943729304 + -1.625 e = -0.943729304 + -1.625 Combine like terms: -0.943729304 + -1.625 = -2.568729304 e = -2.568729304 Simplifying e = -2.568729304Solution
The solution to the problem is based on the solutions from the subproblems. e = {-0.681270696, -2.568729304}
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