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Simplifying (4n + -3)(4n + -3) = 29n Reorder the terms: (-3 + 4n)(4n + -3) = 29n Reorder the terms: (-3 + 4n)(-3 + 4n) = 29n Multiply (-3 + 4n) * (-3 + 4n) (-3(-3 + 4n) + 4n * (-3 + 4n)) = 29n ((-3 * -3 + 4n * -3) + 4n * (-3 + 4n)) = 29n ((9 + -12n) + 4n * (-3 + 4n)) = 29n (9 + -12n + (-3 * 4n + 4n * 4n)) = 29n (9 + -12n + (-12n + 16n2)) = 29n Combine like terms: -12n + -12n = -24n (9 + -24n + 16n2) = 29n Solving 9 + -24n + 16n2 = 29n Solving for variable 'n'. Reorder the terms: 9 + -24n + -29n + 16n2 = 29n + -29n Combine like terms: -24n + -29n = -53n 9 + -53n + 16n2 = 29n + -29n Combine like terms: 29n + -29n = 0 9 + -53n + 16n2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 0.5625 + -3.3125n + n2 = 0 Move the constant term to the right: Add '-0.5625' to each side of the equation. 0.5625 + -3.3125n + -0.5625 + n2 = 0 + -0.5625 Reorder the terms: 0.5625 + -0.5625 + -3.3125n + n2 = 0 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + -3.3125n + n2 = 0 + -0.5625 -3.3125n + n2 = 0 + -0.5625 Combine like terms: 0 + -0.5625 = -0.5625 -3.3125n + n2 = -0.5625 The n term is -3.3125n. Take half its coefficient (-1.65625). Square it (2.743164063) and add it to both sides. Add '2.743164063' to each side of the equation. -3.3125n + 2.743164063 + n2 = -0.5625 + 2.743164063 Reorder the terms: 2.743164063 + -3.3125n + n2 = -0.5625 + 2.743164063 Combine like terms: -0.5625 + 2.743164063 = 2.180664063 2.743164063 + -3.3125n + n2 = 2.180664063 Factor a perfect square on the left side: (n + -1.65625)(n + -1.65625) = 2.180664063 Calculate the square root of the right side: 1.476707169 Break this problem into two subproblems by setting (n + -1.65625) equal to 1.476707169 and -1.476707169.Subproblem 1
n + -1.65625 = 1.476707169 Simplifying n + -1.65625 = 1.476707169 Reorder the terms: -1.65625 + n = 1.476707169 Solving -1.65625 + n = 1.476707169 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.65625' to each side of the equation. -1.65625 + 1.65625 + n = 1.476707169 + 1.65625 Combine like terms: -1.65625 + 1.65625 = 0.00000 0.00000 + n = 1.476707169 + 1.65625 n = 1.476707169 + 1.65625 Combine like terms: 1.476707169 + 1.65625 = 3.132957169 n = 3.132957169 Simplifying n = 3.132957169Subproblem 2
n + -1.65625 = -1.476707169 Simplifying n + -1.65625 = -1.476707169 Reorder the terms: -1.65625 + n = -1.476707169 Solving -1.65625 + n = -1.476707169 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.65625' to each side of the equation. -1.65625 + 1.65625 + n = -1.476707169 + 1.65625 Combine like terms: -1.65625 + 1.65625 = 0.00000 0.00000 + n = -1.476707169 + 1.65625 n = -1.476707169 + 1.65625 Combine like terms: -1.476707169 + 1.65625 = 0.179542831 n = 0.179542831 Simplifying n = 0.179542831Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.132957169, 0.179542831}
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