4n(n-3)+7=

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Solution for 4n(n-3)+7= equation:


Simplifying
4n(n + -3) + 7 = 0

Reorder the terms:
4n(-3 + n) + 7 = 0
(-3 * 4n + n * 4n) + 7 = 0
(-12n + 4n2) + 7 = 0

Reorder the terms:
7 + -12n + 4n2 = 0

Solving
7 + -12n + 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'.
1.75 + -3n + n2 = 0

Move the constant term to the right:

Add '-1.75' to each side of the equation.
1.75 + -3n + -1.75 + n2 = 0 + -1.75

Reorder the terms:
1.75 + -1.75 + -3n + n2 = 0 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + -3n + n2 = 0 + -1.75
-3n + n2 = 0 + -1.75

Combine like terms: 0 + -1.75 = -1.75
-3n + n2 = -1.75

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = -1.75 + 2.25

Reorder the terms:
2.25 + -3n + n2 = -1.75 + 2.25

Combine like terms: -1.75 + 2.25 = 0.5
2.25 + -3n + n2 = 0.5

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(n + -1.5) equal to 0.707106781 and -0.707106781.

Subproblem 1

n + -1.5 = 0.707106781 Simplifying n + -1.5 = 0.707106781 Reorder the terms: -1.5 + n = 0.707106781 Solving -1.5 + n = 0.707106781 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 0.707106781 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 0.707106781 + 1.5 n = 0.707106781 + 1.5 Combine like terms: 0.707106781 + 1.5 = 2.207106781 n = 2.207106781 Simplifying n = 2.207106781

Subproblem 2

n + -1.5 = -0.707106781 Simplifying n + -1.5 = -0.707106781 Reorder the terms: -1.5 + n = -0.707106781 Solving -1.5 + n = -0.707106781 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -0.707106781 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -0.707106781 + 1.5 n = -0.707106781 + 1.5 Combine like terms: -0.707106781 + 1.5 = 0.792893219 n = 0.792893219 Simplifying n = 0.792893219

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.207106781, 0.792893219}

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