n^2=4n+10

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Solution for n^2=4n+10 equation:


Simplifying
n2 = 4n + 10

Reorder the terms:
n2 = 10 + 4n

Solving
n2 = 10 + 4n

Solving for variable 'n'.

Reorder the terms:
-10 + -4n + n2 = 10 + 4n + -10 + -4n

Reorder the terms:
-10 + -4n + n2 = 10 + -10 + 4n + -4n

Combine like terms: 10 + -10 = 0
-10 + -4n + n2 = 0 + 4n + -4n
-10 + -4n + n2 = 4n + -4n

Combine like terms: 4n + -4n = 0
-10 + -4n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -4n + 10 + n2 = 0 + 10

Reorder the terms:
-10 + 10 + -4n + n2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -4n + n2 = 0 + 10
-4n + n2 = 0 + 10

Combine like terms: 0 + 10 = 10
-4n + n2 = 10

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = 10 + 4

Reorder the terms:
4 + -4n + n2 = 10 + 4

Combine like terms: 10 + 4 = 14
4 + -4n + n2 = 14

Factor a perfect square on the left side:
(n + -2)(n + -2) = 14

Calculate the square root of the right side: 3.741657387

Break this problem into two subproblems by setting 
(n + -2) equal to 3.741657387 and -3.741657387.

Subproblem 1

n + -2 = 3.741657387 Simplifying n + -2 = 3.741657387 Reorder the terms: -2 + n = 3.741657387 Solving -2 + n = 3.741657387 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 3.741657387 + 2 Combine like terms: -2 + 2 = 0 0 + n = 3.741657387 + 2 n = 3.741657387 + 2 Combine like terms: 3.741657387 + 2 = 5.741657387 n = 5.741657387 Simplifying n = 5.741657387

Subproblem 2

n + -2 = -3.741657387 Simplifying n + -2 = -3.741657387 Reorder the terms: -2 + n = -3.741657387 Solving -2 + n = -3.741657387 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -3.741657387 + 2 Combine like terms: -2 + 2 = 0 0 + n = -3.741657387 + 2 n = -3.741657387 + 2 Combine like terms: -3.741657387 + 2 = -1.741657387 n = -1.741657387 Simplifying n = -1.741657387

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.741657387, -1.741657387}

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