(4n-3)(4n-3)=29n

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Solution for (4n-3)(4n-3)=29n equation:


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 = 3.132957169

Subproblem 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 = 0.179542831

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.132957169, 0.179542831}

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