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Simplifying 2n2 + -2n = 7 Reorder the terms: -2n + 2n2 = 7 Solving -2n + 2n2 = 7 Solving for variable 'n'. Reorder the terms: -7 + -2n + 2n2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + -2n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + -1n + n2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -1n + 3.5 + n2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -1n + n2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -1n + n2 = 0 + 3.5 -1n + n2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -1n + n2 = 3.5 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 3.5 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 3.5 + 0.25 Combine like terms: 3.5 + 0.25 = 3.75 0.25 + -1n + n2 = 3.75 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (n + -0.5) equal to 1.936491673 and -1.936491673.Subproblem 1
n + -0.5 = 1.936491673 Simplifying n + -0.5 = 1.936491673 Reorder the terms: -0.5 + n = 1.936491673 Solving -0.5 + n = 1.936491673 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 1.936491673 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.936491673 + 0.5 n = 1.936491673 + 0.5 Combine like terms: 1.936491673 + 0.5 = 2.436491673 n = 2.436491673 Simplifying n = 2.436491673Subproblem 2
n + -0.5 = -1.936491673 Simplifying n + -0.5 = -1.936491673 Reorder the terms: -0.5 + n = -1.936491673 Solving -0.5 + n = -1.936491673 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -1.936491673 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.936491673 + 0.5 n = -1.936491673 + 0.5 Combine like terms: -1.936491673 + 0.5 = -1.436491673 n = -1.436491673 Simplifying n = -1.436491673Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.436491673, -1.436491673}
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