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Simplifying 7n2 + -5n + -14 = 21 Reorder the terms: -14 + -5n + 7n2 = 21 Solving -14 + -5n + 7n2 = 21 Solving for variable 'n'. Reorder the terms: -14 + -21 + -5n + 7n2 = 21 + -21 Combine like terms: -14 + -21 = -35 -35 + -5n + 7n2 = 21 + -21 Combine like terms: 21 + -21 = 0 -35 + -5n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5 + -0.7142857143n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.7142857143n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.7142857143n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.7142857143n + n2 = 0 + 5 -0.7142857143n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.7142857143n + n2 = 5 The n term is -0.7142857143n. Take half its coefficient (-0.3571428572). Square it (0.1275510204) and add it to both sides. Add '0.1275510204' to each side of the equation. -0.7142857143n + 0.1275510204 + n2 = 5 + 0.1275510204 Reorder the terms: 0.1275510204 + -0.7142857143n + n2 = 5 + 0.1275510204 Combine like terms: 5 + 0.1275510204 = 5.1275510204 0.1275510204 + -0.7142857143n + n2 = 5.1275510204 Factor a perfect square on the left side: (n + -0.3571428572)(n + -0.3571428572) = 5.1275510204 Calculate the square root of the right side: 2.264409641 Break this problem into two subproblems by setting (n + -0.3571428572) equal to 2.264409641 and -2.264409641.Subproblem 1
n + -0.3571428572 = 2.264409641 Simplifying n + -0.3571428572 = 2.264409641 Reorder the terms: -0.3571428572 + n = 2.264409641 Solving -0.3571428572 + n = 2.264409641 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + n = 2.264409641 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + n = 2.264409641 + 0.3571428572 n = 2.264409641 + 0.3571428572 Combine like terms: 2.264409641 + 0.3571428572 = 2.6215524982 n = 2.6215524982 Simplifying n = 2.6215524982Subproblem 2
n + -0.3571428572 = -2.264409641 Simplifying n + -0.3571428572 = -2.264409641 Reorder the terms: -0.3571428572 + n = -2.264409641 Solving -0.3571428572 + n = -2.264409641 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + n = -2.264409641 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + n = -2.264409641 + 0.3571428572 n = -2.264409641 + 0.3571428572 Combine like terms: -2.264409641 + 0.3571428572 = -1.9072667838 n = -1.9072667838 Simplifying n = -1.9072667838Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.6215524982, -1.9072667838}
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