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Simplifying 6n2 + 7n = n2 + 1 Reorder the terms: 7n + 6n2 = n2 + 1 Reorder the terms: 7n + 6n2 = 1 + n2 Solving 7n + 6n2 = 1 + n2 Solving for variable 'n'. Reorder the terms: -1 + 7n + 6n2 + -1n2 = 1 + n2 + -1 + -1n2 Combine like terms: 6n2 + -1n2 = 5n2 -1 + 7n + 5n2 = 1 + n2 + -1 + -1n2 Reorder the terms: -1 + 7n + 5n2 = 1 + -1 + n2 + -1n2 Combine like terms: 1 + -1 = 0 -1 + 7n + 5n2 = 0 + n2 + -1n2 -1 + 7n + 5n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 -1 + 7n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.2 + 1.4n + n2 = 0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + 1.4n + 0.2 + n2 = 0 + 0.2 Reorder the terms: -0.2 + 0.2 + 1.4n + n2 = 0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + 1.4n + n2 = 0 + 0.2 1.4n + n2 = 0 + 0.2 Combine like terms: 0 + 0.2 = 0.2 1.4n + n2 = 0.2 The n term is 1.4n. Take half its coefficient (0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. 1.4n + 0.49 + n2 = 0.2 + 0.49 Reorder the terms: 0.49 + 1.4n + n2 = 0.2 + 0.49 Combine like terms: 0.2 + 0.49 = 0.69 0.49 + 1.4n + n2 = 0.69 Factor a perfect square on the left side: (n + 0.7)(n + 0.7) = 0.69 Calculate the square root of the right side: 0.830662386 Break this problem into two subproblems by setting (n + 0.7) equal to 0.830662386 and -0.830662386.Subproblem 1
n + 0.7 = 0.830662386 Simplifying n + 0.7 = 0.830662386 Reorder the terms: 0.7 + n = 0.830662386 Solving 0.7 + n = 0.830662386 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = 0.830662386 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = 0.830662386 + -0.7 n = 0.830662386 + -0.7 Combine like terms: 0.830662386 + -0.7 = 0.130662386 n = 0.130662386 Simplifying n = 0.130662386Subproblem 2
n + 0.7 = -0.830662386 Simplifying n + 0.7 = -0.830662386 Reorder the terms: 0.7 + n = -0.830662386 Solving 0.7 + n = -0.830662386 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = -0.830662386 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = -0.830662386 + -0.7 n = -0.830662386 + -0.7 Combine like terms: -0.830662386 + -0.7 = -1.530662386 n = -1.530662386 Simplifying n = -1.530662386Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.130662386, -1.530662386}
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