4n^2-15n+12=0

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Solution for 4n^2-15n+12=0 equation:


Simplifying
4n2 + -15n + 12 = 0

Reorder the terms:
12 + -15n + 4n2 = 0

Solving
12 + -15n + 4n2 = 0

Solving for variable 'n'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
3 + -3.75n + n2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -3.75n + -3 + n2 = 0 + -3

Reorder the terms:
3 + -3 + -3.75n + n2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -3.75n + n2 = 0 + -3
-3.75n + n2 = 0 + -3

Combine like terms: 0 + -3 = -3
-3.75n + n2 = -3

The n term is -3.75n.  Take half its coefficient (-1.875).
Square it (3.515625) and add it to both sides.

Add '3.515625' to each side of the equation.
-3.75n + 3.515625 + n2 = -3 + 3.515625

Reorder the terms:
3.515625 + -3.75n + n2 = -3 + 3.515625

Combine like terms: -3 + 3.515625 = 0.515625
3.515625 + -3.75n + n2 = 0.515625

Factor a perfect square on the left side:
(n + -1.875)(n + -1.875) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(n + -1.875) equal to 0.718070331 and -0.718070331.

Subproblem 1

n + -1.875 = 0.718070331 Simplifying n + -1.875 = 0.718070331 Reorder the terms: -1.875 + n = 0.718070331 Solving -1.875 + n = 0.718070331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + n = 0.718070331 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + n = 0.718070331 + 1.875 n = 0.718070331 + 1.875 Combine like terms: 0.718070331 + 1.875 = 2.593070331 n = 2.593070331 Simplifying n = 2.593070331

Subproblem 2

n + -1.875 = -0.718070331 Simplifying n + -1.875 = -0.718070331 Reorder the terms: -1.875 + n = -0.718070331 Solving -1.875 + n = -0.718070331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + n = -0.718070331 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + n = -0.718070331 + 1.875 n = -0.718070331 + 1.875 Combine like terms: -0.718070331 + 1.875 = 1.156929669 n = 1.156929669 Simplifying n = 1.156929669

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.593070331, 1.156929669}

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