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Simplifying 7n2 + 2n + -5 = 3n2 Reorder the terms: -5 + 2n + 7n2 = 3n2 Solving -5 + 2n + 7n2 = 3n2 Solving for variable 'n'. Combine like terms: 7n2 + -3n2 = 4n2 -5 + 2n + 4n2 = 3n2 + -3n2 Combine like terms: 3n2 + -3n2 = 0 -5 + 2n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.25 + 0.5n + n2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + 0.5n + 1.25 + n2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + 0.5n + n2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + 0.5n + n2 = 0 + 1.25 0.5n + n2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 0.5n + n2 = 1.25 The n term is 0.5n. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5n + 0.0625 + n2 = 1.25 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 1.25 + 0.0625 Combine like terms: 1.25 + 0.0625 = 1.3125 0.0625 + 0.5n + n2 = 1.3125 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (n + 0.25) equal to 1.145643924 and -1.145643924.Subproblem 1
n + 0.25 = 1.145643924 Simplifying n + 0.25 = 1.145643924 Reorder the terms: 0.25 + n = 1.145643924 Solving 0.25 + n = 1.145643924 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 1.145643924 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.145643924 + -0.25 n = 1.145643924 + -0.25 Combine like terms: 1.145643924 + -0.25 = 0.895643924 n = 0.895643924 Simplifying n = 0.895643924Subproblem 2
n + 0.25 = -1.145643924 Simplifying n + 0.25 = -1.145643924 Reorder the terms: 0.25 + n = -1.145643924 Solving 0.25 + n = -1.145643924 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -1.145643924 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.145643924 + -0.25 n = -1.145643924 + -0.25 Combine like terms: -1.145643924 + -0.25 = -1.395643924 n = -1.395643924 Simplifying n = -1.395643924Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.895643924, -1.395643924}
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