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Simplifying 7n2 + -4n + -14 = 21 Reorder the terms: -14 + -4n + 7n2 = 21 Solving -14 + -4n + 7n2 = 21 Solving for variable 'n'. Reorder the terms: -14 + -21 + -4n + 7n2 = 21 + -21 Combine like terms: -14 + -21 = -35 -35 + -4n + 7n2 = 21 + -21 Combine like terms: 21 + -21 = 0 -35 + -4n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5 + -0.5714285714n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.5714285714n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.5714285714n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.5714285714n + n2 = 0 + 5 -0.5714285714n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.5714285714n + n2 = 5 The n term is -0.5714285714n. Take half its coefficient (-0.2857142857). Square it (0.08163265305) and add it to both sides. Add '0.08163265305' to each side of the equation. -0.5714285714n + 0.08163265305 + n2 = 5 + 0.08163265305 Reorder the terms: 0.08163265305 + -0.5714285714n + n2 = 5 + 0.08163265305 Combine like terms: 5 + 0.08163265305 = 5.08163265305 0.08163265305 + -0.5714285714n + n2 = 5.08163265305 Factor a perfect square on the left side: (n + -0.2857142857)(n + -0.2857142857) = 5.08163265305 Calculate the square root of the right side: 2.254247691 Break this problem into two subproblems by setting (n + -0.2857142857) equal to 2.254247691 and -2.254247691.Subproblem 1
n + -0.2857142857 = 2.254247691 Simplifying n + -0.2857142857 = 2.254247691 Reorder the terms: -0.2857142857 + n = 2.254247691 Solving -0.2857142857 + n = 2.254247691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + n = 2.254247691 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + n = 2.254247691 + 0.2857142857 n = 2.254247691 + 0.2857142857 Combine like terms: 2.254247691 + 0.2857142857 = 2.5399619767 n = 2.5399619767 Simplifying n = 2.5399619767Subproblem 2
n + -0.2857142857 = -2.254247691 Simplifying n + -0.2857142857 = -2.254247691 Reorder the terms: -0.2857142857 + n = -2.254247691 Solving -0.2857142857 + n = -2.254247691 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + n = -2.254247691 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + n = -2.254247691 + 0.2857142857 n = -2.254247691 + 0.2857142857 Combine like terms: -2.254247691 + 0.2857142857 = -1.9685334053 n = -1.9685334053 Simplifying n = -1.9685334053Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.5399619767, -1.9685334053}
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