4n^2+10n+3=0

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Solution for 4n^2+10n+3=0 equation:


Simplifying
4n2 + 10n + 3 = 0

Reorder the terms:
3 + 10n + 4n2 = 0

Solving
3 + 10n + 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'.
0.75 + 2.5n + n2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 2.5n + -0.75 + n2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 2.5n + n2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 2.5n + n2 = 0 + -0.75
2.5n + n2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
2.5n + n2 = -0.75

The n term is 2.5n.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5n + 1.5625 + n2 = -0.75 + 1.5625

Reorder the terms:
1.5625 + 2.5n + n2 = -0.75 + 1.5625

Combine like terms: -0.75 + 1.5625 = 0.8125
1.5625 + 2.5n + n2 = 0.8125

Factor a perfect square on the left side:
(n + 1.25)(n + 1.25) = 0.8125

Calculate the square root of the right side: 0.901387819

Break this problem into two subproblems by setting 
(n + 1.25) equal to 0.901387819 and -0.901387819.

Subproblem 1

n + 1.25 = 0.901387819 Simplifying n + 1.25 = 0.901387819 Reorder the terms: 1.25 + n = 0.901387819 Solving 1.25 + n = 0.901387819 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = 0.901387819 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 0.901387819 + -1.25 n = 0.901387819 + -1.25 Combine like terms: 0.901387819 + -1.25 = -0.348612181 n = -0.348612181 Simplifying n = -0.348612181

Subproblem 2

n + 1.25 = -0.901387819 Simplifying n + 1.25 = -0.901387819 Reorder the terms: 1.25 + n = -0.901387819 Solving 1.25 + n = -0.901387819 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = -0.901387819 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -0.901387819 + -1.25 n = -0.901387819 + -1.25 Combine like terms: -0.901387819 + -1.25 = -2.151387819 n = -2.151387819 Simplifying n = -2.151387819

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.348612181, -2.151387819}

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