5n^2=1-2n

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Solution for 5n^2=1-2n equation:


Simplifying
5n2 = 1 + -2n

Solving
5n2 = 1 + -2n

Solving for variable 'n'.

Reorder the terms:
-1 + 2n + 5n2 = 1 + -2n + -1 + 2n

Reorder the terms:
-1 + 2n + 5n2 = 1 + -1 + -2n + 2n

Combine like terms: 1 + -1 = 0
-1 + 2n + 5n2 = 0 + -2n + 2n
-1 + 2n + 5n2 = -2n + 2n

Combine like terms: -2n + 2n = 0
-1 + 2n + 5n2 = 0

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

Divide each side by '5'.
-0.2 + 0.4n + n2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + 0.4n + 0.2 + n2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + 0.4n + n2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + 0.4n + n2 = 0 + 0.2
0.4n + n2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
0.4n + n2 = 0.2

The n term is 0.4n.  Take half its coefficient (0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
0.4n + 0.04 + n2 = 0.2 + 0.04

Reorder the terms:
0.04 + 0.4n + n2 = 0.2 + 0.04

Combine like terms: 0.2 + 0.04 = 0.24
0.04 + 0.4n + n2 = 0.24

Factor a perfect square on the left side:
(n + 0.2)(n + 0.2) = 0.24

Calculate the square root of the right side: 0.489897949

Break this problem into two subproblems by setting 
(n + 0.2) equal to 0.489897949 and -0.489897949.

Subproblem 1

n + 0.2 = 0.489897949 Simplifying n + 0.2 = 0.489897949 Reorder the terms: 0.2 + n = 0.489897949 Solving 0.2 + n = 0.489897949 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + n = 0.489897949 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + n = 0.489897949 + -0.2 n = 0.489897949 + -0.2 Combine like terms: 0.489897949 + -0.2 = 0.289897949 n = 0.289897949 Simplifying n = 0.289897949

Subproblem 2

n + 0.2 = -0.489897949 Simplifying n + 0.2 = -0.489897949 Reorder the terms: 0.2 + n = -0.489897949 Solving 0.2 + n = -0.489897949 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + n = -0.489897949 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + n = -0.489897949 + -0.2 n = -0.489897949 + -0.2 Combine like terms: -0.489897949 + -0.2 = -0.689897949 n = -0.689897949 Simplifying n = -0.689897949

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.289897949, -0.689897949}

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