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Simplifying 16n2 + 4 = 19n Reorder the terms: 4 + 16n2 = 19n Solving 4 + 16n2 = 19n Solving for variable 'n'. Reorder the terms: 4 + -19n + 16n2 = 19n + -19n Combine like terms: 19n + -19n = 0 4 + -19n + 16n2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.25 + -1.1875n + n2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -1.1875n + -0.25 + n2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -1.1875n + n2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -1.1875n + n2 = 0 + -0.25 -1.1875n + n2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -1.1875n + n2 = -0.25 The n term is -1.1875n. Take half its coefficient (-0.59375). Square it (0.3525390625) and add it to both sides. Add '0.3525390625' to each side of the equation. -1.1875n + 0.3525390625 + n2 = -0.25 + 0.3525390625 Reorder the terms: 0.3525390625 + -1.1875n + n2 = -0.25 + 0.3525390625 Combine like terms: -0.25 + 0.3525390625 = 0.1025390625 0.3525390625 + -1.1875n + n2 = 0.1025390625 Factor a perfect square on the left side: (n + -0.59375)(n + -0.59375) = 0.1025390625 Calculate the square root of the right side: 0.320217211 Break this problem into two subproblems by setting (n + -0.59375) equal to 0.320217211 and -0.320217211.Subproblem 1
n + -0.59375 = 0.320217211 Simplifying n + -0.59375 = 0.320217211 Reorder the terms: -0.59375 + n = 0.320217211 Solving -0.59375 + n = 0.320217211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.59375' to each side of the equation. -0.59375 + 0.59375 + n = 0.320217211 + 0.59375 Combine like terms: -0.59375 + 0.59375 = 0.00000 0.00000 + n = 0.320217211 + 0.59375 n = 0.320217211 + 0.59375 Combine like terms: 0.320217211 + 0.59375 = 0.913967211 n = 0.913967211 Simplifying n = 0.913967211Subproblem 2
n + -0.59375 = -0.320217211 Simplifying n + -0.59375 = -0.320217211 Reorder the terms: -0.59375 + n = -0.320217211 Solving -0.59375 + n = -0.320217211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.59375' to each side of the equation. -0.59375 + 0.59375 + n = -0.320217211 + 0.59375 Combine like terms: -0.59375 + 0.59375 = 0.00000 0.00000 + n = -0.320217211 + 0.59375 n = -0.320217211 + 0.59375 Combine like terms: -0.320217211 + 0.59375 = 0.273532789 n = 0.273532789 Simplifying n = 0.273532789Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.913967211, 0.273532789}
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