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Simplifying 16n2 + -111n + 14 = 0 Reorder the terms: 14 + -111n + 16n2 = 0 Solving 14 + -111n + 16n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.875 + -6.9375n + n2 = 0 Move the constant term to the right: Add '-0.875' to each side of the equation. 0.875 + -6.9375n + -0.875 + n2 = 0 + -0.875 Reorder the terms: 0.875 + -0.875 + -6.9375n + n2 = 0 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + -6.9375n + n2 = 0 + -0.875 -6.9375n + n2 = 0 + -0.875 Combine like terms: 0 + -0.875 = -0.875 -6.9375n + n2 = -0.875 The n term is -6.9375n. Take half its coefficient (-3.46875). Square it (12.03222656) and add it to both sides. Add '12.03222656' to each side of the equation. -6.9375n + 12.03222656 + n2 = -0.875 + 12.03222656 Reorder the terms: 12.03222656 + -6.9375n + n2 = -0.875 + 12.03222656 Combine like terms: -0.875 + 12.03222656 = 11.15722656 12.03222656 + -6.9375n + n2 = 11.15722656 Factor a perfect square on the left side: (n + -3.46875)(n + -3.46875) = 11.15722656 Calculate the square root of the right side: 3.340243488 Break this problem into two subproblems by setting (n + -3.46875) equal to 3.340243488 and -3.340243488.Subproblem 1
n + -3.46875 = 3.340243488 Simplifying n + -3.46875 = 3.340243488 Reorder the terms: -3.46875 + n = 3.340243488 Solving -3.46875 + n = 3.340243488 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.46875' to each side of the equation. -3.46875 + 3.46875 + n = 3.340243488 + 3.46875 Combine like terms: -3.46875 + 3.46875 = 0.00000 0.00000 + n = 3.340243488 + 3.46875 n = 3.340243488 + 3.46875 Combine like terms: 3.340243488 + 3.46875 = 6.808993488 n = 6.808993488 Simplifying n = 6.808993488Subproblem 2
n + -3.46875 = -3.340243488 Simplifying n + -3.46875 = -3.340243488 Reorder the terms: -3.46875 + n = -3.340243488 Solving -3.46875 + n = -3.340243488 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.46875' to each side of the equation. -3.46875 + 3.46875 + n = -3.340243488 + 3.46875 Combine like terms: -3.46875 + 3.46875 = 0.00000 0.00000 + n = -3.340243488 + 3.46875 n = -3.340243488 + 3.46875 Combine like terms: -3.340243488 + 3.46875 = 0.128506512 n = 0.128506512 Simplifying n = 0.128506512Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.808993488, 0.128506512}
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