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Simplifying 4n2 + 4n + 1 = 20 Reorder the terms: 1 + 4n + 4n2 = 20 Solving 1 + 4n + 4n2 = 20 Solving for variable 'n'. Reorder the terms: 1 + -20 + 4n + 4n2 = 20 + -20 Combine like terms: 1 + -20 = -19 -19 + 4n + 4n2 = 20 + -20 Combine like terms: 20 + -20 = 0 -19 + 4n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -4.75 + n + n2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + n + 4.75 + n2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + n + n2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + n + n2 = 0 + 4.75 n + n2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 n + n2 = 4.75 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + n2 = 4.75 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 4.75 + 0.25 Combine like terms: 4.75 + 0.25 = 5 1.25 + n2 = 5 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 5 Calculate the square root of the right side: 2.236067978 Break this problem into two subproblems by setting (n + 0.5) equal to 2.236067978 and -2.236067978.Subproblem 1
n + 0.5 = 2.236067978 Simplifying n + 0.5 = 2.236067978 Reorder the terms: 0.5 + n = 2.236067978 Solving 0.5 + n = 2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 2.236067978 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 2.236067978 + -0.5 n = 2.236067978 + -0.5 Combine like terms: 2.236067978 + -0.5 = 1.736067978 n = 1.736067978 Simplifying n = 1.736067978Subproblem 2
n + 0.5 = -2.236067978 Simplifying n + 0.5 = -2.236067978 Reorder the terms: 0.5 + n = -2.236067978 Solving 0.5 + n = -2.236067978 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -2.236067978 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -2.236067978 + -0.5 n = -2.236067978 + -0.5 Combine like terms: -2.236067978 + -0.5 = -2.736067978 n = -2.736067978 Simplifying n = -2.736067978Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.736067978, -2.736067978}
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