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Simplifying -7n2 + 57n + 14 = 0 Reorder the terms: 14 + 57n + -7n2 = 0 Solving 14 + 57n + -7n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -2 + -8.142857143n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -8.142857143n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + -8.142857143n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -8.142857143n + n2 = 0 + 2 -8.142857143n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 -8.142857143n + n2 = 2 The n term is -8.142857143n. Take half its coefficient (-4.071428572). Square it (16.57653062) and add it to both sides. Add '16.57653062' to each side of the equation. -8.142857143n + 16.57653062 + n2 = 2 + 16.57653062 Reorder the terms: 16.57653062 + -8.142857143n + n2 = 2 + 16.57653062 Combine like terms: 2 + 16.57653062 = 18.57653062 16.57653062 + -8.142857143n + n2 = 18.57653062 Factor a perfect square on the left side: (n + -4.071428572)(n + -4.071428572) = 18.57653062 Calculate the square root of the right side: 4.310049956 Break this problem into two subproblems by setting (n + -4.071428572) equal to 4.310049956 and -4.310049956.Subproblem 1
n + -4.071428572 = 4.310049956 Simplifying n + -4.071428572 = 4.310049956 Reorder the terms: -4.071428572 + n = 4.310049956 Solving -4.071428572 + n = 4.310049956 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.071428572' to each side of the equation. -4.071428572 + 4.071428572 + n = 4.310049956 + 4.071428572 Combine like terms: -4.071428572 + 4.071428572 = 0.000000000 0.000000000 + n = 4.310049956 + 4.071428572 n = 4.310049956 + 4.071428572 Combine like terms: 4.310049956 + 4.071428572 = 8.381478528 n = 8.381478528 Simplifying n = 8.381478528Subproblem 2
n + -4.071428572 = -4.310049956 Simplifying n + -4.071428572 = -4.310049956 Reorder the terms: -4.071428572 + n = -4.310049956 Solving -4.071428572 + n = -4.310049956 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.071428572' to each side of the equation. -4.071428572 + 4.071428572 + n = -4.310049956 + 4.071428572 Combine like terms: -4.071428572 + 4.071428572 = 0.000000000 0.000000000 + n = -4.310049956 + 4.071428572 n = -4.310049956 + 4.071428572 Combine like terms: -4.310049956 + 4.071428572 = -0.238621384 n = -0.238621384 Simplifying n = -0.238621384Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.381478528, -0.238621384}
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