5*n^2+4*n=10

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Solution for 5*n^2+4*n=10 equation:


Simplifying
5n2 + 4n = 10

Reorder the terms:
4n + 5n2 = 10

Solving
4n + 5n2 = 10

Solving for variable 'n'.

Reorder the terms:
-10 + 4n + 5n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 4n + 5n2 = 0

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

Divide each side by '5'.
-2 + 0.8n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.8n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.8n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.8n + n2 = 0 + 2
0.8n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.8n + n2 = 2

The n term is 0.8n.  Take half its coefficient (0.4).
Square it (0.16) and add it to both sides.

Add '0.16' to each side of the equation.
0.8n + 0.16 + n2 = 2 + 0.16

Reorder the terms:
0.16 + 0.8n + n2 = 2 + 0.16

Combine like terms: 2 + 0.16 = 2.16
0.16 + 0.8n + n2 = 2.16

Factor a perfect square on the left side:
(n + 0.4)(n + 0.4) = 2.16

Calculate the square root of the right side: 1.469693846

Break this problem into two subproblems by setting 
(n + 0.4) equal to 1.469693846 and -1.469693846.

Subproblem 1

n + 0.4 = 1.469693846 Simplifying n + 0.4 = 1.469693846 Reorder the terms: 0.4 + n = 1.469693846 Solving 0.4 + n = 1.469693846 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + n = 1.469693846 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + n = 1.469693846 + -0.4 n = 1.469693846 + -0.4 Combine like terms: 1.469693846 + -0.4 = 1.069693846 n = 1.069693846 Simplifying n = 1.069693846

Subproblem 2

n + 0.4 = -1.469693846 Simplifying n + 0.4 = -1.469693846 Reorder the terms: 0.4 + n = -1.469693846 Solving 0.4 + n = -1.469693846 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + n = -1.469693846 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + n = -1.469693846 + -0.4 n = -1.469693846 + -0.4 Combine like terms: -1.469693846 + -0.4 = -1.869693846 n = -1.869693846 Simplifying n = -1.869693846

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.069693846, -1.869693846}

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