-6n(-6n-5)=7n+30

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Solution for -6n(-6n-5)=7n+30 equation:


Simplifying
-6n(-6n + -5) = 7n + 30

Reorder the terms:
-6n(-5 + -6n) = 7n + 30
(-5 * -6n + -6n * -6n) = 7n + 30
(30n + 36n2) = 7n + 30

Reorder the terms:
30n + 36n2 = 30 + 7n

Solving
30n + 36n2 = 30 + 7n

Solving for variable 'n'.

Reorder the terms:
-30 + 30n + -7n + 36n2 = 30 + 7n + -30 + -7n

Combine like terms: 30n + -7n = 23n
-30 + 23n + 36n2 = 30 + 7n + -30 + -7n

Reorder the terms:
-30 + 23n + 36n2 = 30 + -30 + 7n + -7n

Combine like terms: 30 + -30 = 0
-30 + 23n + 36n2 = 0 + 7n + -7n
-30 + 23n + 36n2 = 7n + -7n

Combine like terms: 7n + -7n = 0
-30 + 23n + 36n2 = 0

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

Divide each side by '36'.
-0.8333333333 + 0.6388888889n + n2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 0.6388888889n + 0.8333333333 + n2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 0.6388888889n + n2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 0.6388888889n + n2 = 0 + 0.8333333333
0.6388888889n + n2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
0.6388888889n + n2 = 0.8333333333

The n term is 0.6388888889n.  Take half its coefficient (0.3194444445).
Square it (0.1020447531) and add it to both sides.

Add '0.1020447531' to each side of the equation.
0.6388888889n + 0.1020447531 + n2 = 0.8333333333 + 0.1020447531

Reorder the terms:
0.1020447531 + 0.6388888889n + n2 = 0.8333333333 + 0.1020447531

Combine like terms: 0.8333333333 + 0.1020447531 = 0.9353780864
0.1020447531 + 0.6388888889n + n2 = 0.9353780864

Factor a perfect square on the left side:
(n + 0.3194444445)(n + 0.3194444445) = 0.9353780864

Calculate the square root of the right side: 0.967149464

Break this problem into two subproblems by setting 
(n + 0.3194444445) equal to 0.967149464 and -0.967149464.

Subproblem 1

n + 0.3194444445 = 0.967149464 Simplifying n + 0.3194444445 = 0.967149464 Reorder the terms: 0.3194444445 + n = 0.967149464 Solving 0.3194444445 + n = 0.967149464 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3194444445' to each side of the equation. 0.3194444445 + -0.3194444445 + n = 0.967149464 + -0.3194444445 Combine like terms: 0.3194444445 + -0.3194444445 = 0.0000000000 0.0000000000 + n = 0.967149464 + -0.3194444445 n = 0.967149464 + -0.3194444445 Combine like terms: 0.967149464 + -0.3194444445 = 0.6477050195 n = 0.6477050195 Simplifying n = 0.6477050195

Subproblem 2

n + 0.3194444445 = -0.967149464 Simplifying n + 0.3194444445 = -0.967149464 Reorder the terms: 0.3194444445 + n = -0.967149464 Solving 0.3194444445 + n = -0.967149464 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3194444445' to each side of the equation. 0.3194444445 + -0.3194444445 + n = -0.967149464 + -0.3194444445 Combine like terms: 0.3194444445 + -0.3194444445 = 0.0000000000 0.0000000000 + n = -0.967149464 + -0.3194444445 n = -0.967149464 + -0.3194444445 Combine like terms: -0.967149464 + -0.3194444445 = -1.2865939085 n = -1.2865939085 Simplifying n = -1.2865939085

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.6477050195, -1.2865939085}

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