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Simplifying 7n2 + -6n + -14 = 21 Reorder the terms: -14 + -6n + 7n2 = 21 Solving -14 + -6n + 7n2 = 21 Solving for variable 'n'. Reorder the terms: -14 + -21 + -6n + 7n2 = 21 + -21 Combine like terms: -14 + -21 = -35 -35 + -6n + 7n2 = 21 + -21 Combine like terms: 21 + -21 = 0 -35 + -6n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5 + -0.8571428571n + n2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.8571428571n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + -0.8571428571n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.8571428571n + n2 = 0 + 5 -0.8571428571n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.8571428571n + n2 = 5 The n term is -0.8571428571n. Take half its coefficient (-0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. -0.8571428571n + 0.1836734694 + n2 = 5 + 0.1836734694 Reorder the terms: 0.1836734694 + -0.8571428571n + n2 = 5 + 0.1836734694 Combine like terms: 5 + 0.1836734694 = 5.1836734694 0.1836734694 + -0.8571428571n + n2 = 5.1836734694 Factor a perfect square on the left side: (n + -0.4285714286)(n + -0.4285714286) = 5.1836734694 Calculate the square root of the right side: 2.276768207 Break this problem into two subproblems by setting (n + -0.4285714286) equal to 2.276768207 and -2.276768207.Subproblem 1
n + -0.4285714286 = 2.276768207 Simplifying n + -0.4285714286 = 2.276768207 Reorder the terms: -0.4285714286 + n = 2.276768207 Solving -0.4285714286 + n = 2.276768207 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + n = 2.276768207 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + n = 2.276768207 + 0.4285714286 n = 2.276768207 + 0.4285714286 Combine like terms: 2.276768207 + 0.4285714286 = 2.7053396356 n = 2.7053396356 Simplifying n = 2.7053396356Subproblem 2
n + -0.4285714286 = -2.276768207 Simplifying n + -0.4285714286 = -2.276768207 Reorder the terms: -0.4285714286 + n = -2.276768207 Solving -0.4285714286 + n = -2.276768207 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + n = -2.276768207 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + n = -2.276768207 + 0.4285714286 n = -2.276768207 + 0.4285714286 Combine like terms: -2.276768207 + 0.4285714286 = -1.8481967784 n = -1.8481967784 Simplifying n = -1.8481967784Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.7053396356, -1.8481967784}
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