4n^2=7n-2

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Solution for 4n^2=7n-2 equation:


Simplifying
4n2 = 7n + -2

Reorder the terms:
4n2 = -2 + 7n

Solving
4n2 = -2 + 7n

Solving for variable 'n'.

Reorder the terms:
2 + -7n + 4n2 = -2 + 7n + 2 + -7n

Reorder the terms:
2 + -7n + 4n2 = -2 + 2 + 7n + -7n

Combine like terms: -2 + 2 = 0
2 + -7n + 4n2 = 0 + 7n + -7n
2 + -7n + 4n2 = 7n + -7n

Combine like terms: 7n + -7n = 0
2 + -7n + 4n2 = 0

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

Divide each side by '4'.
0.5 + -1.75n + n2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -1.75n + -0.5 + n2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -1.75n + n2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -1.75n + n2 = 0 + -0.5
-1.75n + n2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-1.75n + n2 = -0.5

The n term is -1.75n.  Take half its coefficient (-0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
-1.75n + 0.765625 + n2 = -0.5 + 0.765625

Reorder the terms:
0.765625 + -1.75n + n2 = -0.5 + 0.765625

Combine like terms: -0.5 + 0.765625 = 0.265625
0.765625 + -1.75n + n2 = 0.265625

Factor a perfect square on the left side:
(n + -0.875)(n + -0.875) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(n + -0.875) equal to 0.515388203 and -0.515388203.

Subproblem 1

n + -0.875 = 0.515388203 Simplifying n + -0.875 = 0.515388203 Reorder the terms: -0.875 + n = 0.515388203 Solving -0.875 + n = 0.515388203 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + n = 0.515388203 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + n = 0.515388203 + 0.875 n = 0.515388203 + 0.875 Combine like terms: 0.515388203 + 0.875 = 1.390388203 n = 1.390388203 Simplifying n = 1.390388203

Subproblem 2

n + -0.875 = -0.515388203 Simplifying n + -0.875 = -0.515388203 Reorder the terms: -0.875 + n = -0.515388203 Solving -0.875 + n = -0.515388203 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + n = -0.515388203 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + n = -0.515388203 + 0.875 n = -0.515388203 + 0.875 Combine like terms: -0.515388203 + 0.875 = 0.359611797 n = 0.359611797 Simplifying n = 0.359611797

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.390388203, 0.359611797}

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