-4n(5+5n)-1=-21-7n

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Solution for -4n(5+5n)-1=-21-7n equation:


Simplifying
-4n(5 + 5n) + -1 = -21 + -7n
(5 * -4n + 5n * -4n) + -1 = -21 + -7n
(-20n + -20n2) + -1 = -21 + -7n

Reorder the terms:
-1 + -20n + -20n2 = -21 + -7n

Solving
-1 + -20n + -20n2 = -21 + -7n

Solving for variable 'n'.

Reorder the terms:
-1 + 21 + -20n + 7n + -20n2 = -21 + -7n + 21 + 7n

Combine like terms: -1 + 21 = 20
20 + -20n + 7n + -20n2 = -21 + -7n + 21 + 7n

Combine like terms: -20n + 7n = -13n
20 + -13n + -20n2 = -21 + -7n + 21 + 7n

Reorder the terms:
20 + -13n + -20n2 = -21 + 21 + -7n + 7n

Combine like terms: -21 + 21 = 0
20 + -13n + -20n2 = 0 + -7n + 7n
20 + -13n + -20n2 = -7n + 7n

Combine like terms: -7n + 7n = 0
20 + -13n + -20n2 = 0

Begin completing the square.  Divide all terms by
-20 the coefficient of the squared term: 

Divide each side by '-20'.
-1 + 0.65n + n2 = 0.0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.65n + 1 + n2 = 0.0 + 1

Reorder the terms:
-1 + 1 + 0.65n + n2 = 0.0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.65n + n2 = 0.0 + 1
0.65n + n2 = 0.0 + 1

Combine like terms: 0.0 + 1 = 1
0.65n + n2 = 1

The n term is 0.65n.  Take half its coefficient (0.325).
Square it (0.105625) and add it to both sides.

Add '0.105625' to each side of the equation.
0.65n + 0.105625 + n2 = 1 + 0.105625

Reorder the terms:
0.105625 + 0.65n + n2 = 1 + 0.105625

Combine like terms: 1 + 0.105625 = 1.105625
0.105625 + 0.65n + n2 = 1.105625

Factor a perfect square on the left side:
(n + 0.325)(n + 0.325) = 1.105625

Calculate the square root of the right side: 1.051487042

Break this problem into two subproblems by setting 
(n + 0.325) equal to 1.051487042 and -1.051487042.

Subproblem 1

n + 0.325 = 1.051487042 Simplifying n + 0.325 = 1.051487042 Reorder the terms: 0.325 + n = 1.051487042 Solving 0.325 + n = 1.051487042 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.325' to each side of the equation. 0.325 + -0.325 + n = 1.051487042 + -0.325 Combine like terms: 0.325 + -0.325 = 0.000 0.000 + n = 1.051487042 + -0.325 n = 1.051487042 + -0.325 Combine like terms: 1.051487042 + -0.325 = 0.726487042 n = 0.726487042 Simplifying n = 0.726487042

Subproblem 2

n + 0.325 = -1.051487042 Simplifying n + 0.325 = -1.051487042 Reorder the terms: 0.325 + n = -1.051487042 Solving 0.325 + n = -1.051487042 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.325' to each side of the equation. 0.325 + -0.325 + n = -1.051487042 + -0.325 Combine like terms: 0.325 + -0.325 = 0.000 0.000 + n = -1.051487042 + -0.325 n = -1.051487042 + -0.325 Combine like terms: -1.051487042 + -0.325 = -1.376487042 n = -1.376487042 Simplifying n = -1.376487042

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.726487042, -1.376487042}

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