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Simplifying 68 + -4n = n * n Multiply n * n 68 + -4n = n2 Solving 68 + -4n = n2 Solving for variable 'n'. Combine like terms: n2 + -1n2 = 0 68 + -4n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -68 + 4n + n2 = 0 Move the constant term to the right: Add '68' to each side of the equation. -68 + 4n + 68 + n2 = 0 + 68 Reorder the terms: -68 + 68 + 4n + n2 = 0 + 68 Combine like terms: -68 + 68 = 0 0 + 4n + n2 = 0 + 68 4n + n2 = 0 + 68 Combine like terms: 0 + 68 = 68 4n + n2 = 68 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = 68 + 4 Reorder the terms: 4 + 4n + n2 = 68 + 4 Combine like terms: 68 + 4 = 72 4 + 4n + n2 = 72 Factor a perfect square on the left side: (n + 2)(n + 2) = 72 Calculate the square root of the right side: 8.485281374 Break this problem into two subproblems by setting (n + 2) equal to 8.485281374 and -8.485281374.Subproblem 1
n + 2 = 8.485281374 Simplifying n + 2 = 8.485281374 Reorder the terms: 2 + n = 8.485281374 Solving 2 + n = 8.485281374 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 8.485281374 + -2 Combine like terms: 2 + -2 = 0 0 + n = 8.485281374 + -2 n = 8.485281374 + -2 Combine like terms: 8.485281374 + -2 = 6.485281374 n = 6.485281374 Simplifying n = 6.485281374Subproblem 2
n + 2 = -8.485281374 Simplifying n + 2 = -8.485281374 Reorder the terms: 2 + n = -8.485281374 Solving 2 + n = -8.485281374 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -8.485281374 + -2 Combine like terms: 2 + -2 = 0 0 + n = -8.485281374 + -2 n = -8.485281374 + -2 Combine like terms: -8.485281374 + -2 = -10.485281374 n = -10.485281374 Simplifying n = -10.485281374Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.485281374, -10.485281374}
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