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Simplifying 2n2 + -3n = 7 Reorder the terms: -3n + 2n2 = 7 Solving -3n + 2n2 = 7 Solving for variable 'n'. Reorder the terms: -7 + -3n + 2n2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + -3n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + -1.5n + n2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -1.5n + 3.5 + n2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -1.5n + n2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -1.5n + n2 = 0 + 3.5 -1.5n + n2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -1.5n + n2 = 3.5 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = 3.5 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 3.5 + 0.5625 Combine like terms: 3.5 + 0.5625 = 4.0625 0.5625 + -1.5n + n2 = 4.0625 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (n + -0.75) equal to 2.015564437 and -2.015564437.Subproblem 1
n + -0.75 = 2.015564437 Simplifying n + -0.75 = 2.015564437 Reorder the terms: -0.75 + n = 2.015564437 Solving -0.75 + n = 2.015564437 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 2.015564437 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 2.015564437 + 0.75 n = 2.015564437 + 0.75 Combine like terms: 2.015564437 + 0.75 = 2.765564437 n = 2.765564437 Simplifying n = 2.765564437Subproblem 2
n + -0.75 = -2.015564437 Simplifying n + -0.75 = -2.015564437 Reorder the terms: -0.75 + n = -2.015564437 Solving -0.75 + n = -2.015564437 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -2.015564437 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -2.015564437 + 0.75 n = -2.015564437 + 0.75 Combine like terms: -2.015564437 + 0.75 = -1.265564437 n = -1.265564437 Simplifying n = -1.265564437Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.765564437, -1.265564437}
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