3n^2-22n+7=4

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Solution for 3n^2-22n+7=4 equation:


Simplifying
3n2 + -22n + 7 = 4

Reorder the terms:
7 + -22n + 3n2 = 4

Solving
7 + -22n + 3n2 = 4

Solving for variable 'n'.

Reorder the terms:
7 + -4 + -22n + 3n2 = 4 + -4

Combine like terms: 7 + -4 = 3
3 + -22n + 3n2 = 4 + -4

Combine like terms: 4 + -4 = 0
3 + -22n + 3n2 = 0

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

Divide each side by '3'.
1 + -7.333333333n + n2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -7.333333333n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + -7.333333333n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -7.333333333n + n2 = 0 + -1
-7.333333333n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
-7.333333333n + n2 = -1

The n term is -7.333333333n.  Take half its coefficient (-3.666666667).
Square it (13.44444445) and add it to both sides.

Add '13.44444445' to each side of the equation.
-7.333333333n + 13.44444445 + n2 = -1 + 13.44444445

Reorder the terms:
13.44444445 + -7.333333333n + n2 = -1 + 13.44444445

Combine like terms: -1 + 13.44444445 = 12.44444445
13.44444445 + -7.333333333n + n2 = 12.44444445

Factor a perfect square on the left side:
(n + -3.666666667)(n + -3.666666667) = 12.44444445

Calculate the square root of the right side: 3.527668416

Break this problem into two subproblems by setting 
(n + -3.666666667) equal to 3.527668416 and -3.527668416.

Subproblem 1

n + -3.666666667 = 3.527668416 Simplifying n + -3.666666667 = 3.527668416 Reorder the terms: -3.666666667 + n = 3.527668416 Solving -3.666666667 + n = 3.527668416 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.666666667' to each side of the equation. -3.666666667 + 3.666666667 + n = 3.527668416 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + n = 3.527668416 + 3.666666667 n = 3.527668416 + 3.666666667 Combine like terms: 3.527668416 + 3.666666667 = 7.194335083 n = 7.194335083 Simplifying n = 7.194335083

Subproblem 2

n + -3.666666667 = -3.527668416 Simplifying n + -3.666666667 = -3.527668416 Reorder the terms: -3.666666667 + n = -3.527668416 Solving -3.666666667 + n = -3.527668416 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.666666667' to each side of the equation. -3.666666667 + 3.666666667 + n = -3.527668416 + 3.666666667 Combine like terms: -3.666666667 + 3.666666667 = 0.000000000 0.000000000 + n = -3.527668416 + 3.666666667 n = -3.527668416 + 3.666666667 Combine like terms: -3.527668416 + 3.666666667 = 0.138998251 n = 0.138998251 Simplifying n = 0.138998251

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.194335083, 0.138998251}

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