(6*70)-(6*4)=n*(70-n)

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Solution for (6*70)-(6*4)=n*(70-n) equation:


Simplifying
(6 * 70) + -1(6 * 4) = n(70 + -1n)

Multiply 6 * 70
(420) + -1(6 * 4) = n(70 + -1n)
420 + -1(6 * 4) = n(70 + -1n)

Multiply 6 * 4
420 + -1(24) = n(70 + -1n)

Multiply -1 * 24
420 + -24 = n(70 + -1n)

Combine like terms: 420 + -24 = 396
396 = n(70 + -1n)
396 = (70 * n + -1n * n)
396 = (70n + -1n2)

Solving
396 = 70n + -1n2

Solving for variable 'n'.

Reorder the terms:
396 + -70n + n2 = 70n + -70n + -1n2 + n2

Combine like terms: 70n + -70n = 0
396 + -70n + n2 = 0 + -1n2 + n2
396 + -70n + n2 = -1n2 + n2

Combine like terms: -1n2 + n2 = 0
396 + -70n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-396' to each side of the equation.
396 + -70n + -396 + n2 = 0 + -396

Reorder the terms:
396 + -396 + -70n + n2 = 0 + -396

Combine like terms: 396 + -396 = 0
0 + -70n + n2 = 0 + -396
-70n + n2 = 0 + -396

Combine like terms: 0 + -396 = -396
-70n + n2 = -396

The n term is -70n.  Take half its coefficient (-35).
Square it (1225) and add it to both sides.

Add '1225' to each side of the equation.
-70n + 1225 + n2 = -396 + 1225

Reorder the terms:
1225 + -70n + n2 = -396 + 1225

Combine like terms: -396 + 1225 = 829
1225 + -70n + n2 = 829

Factor a perfect square on the left side:
(n + -35)(n + -35) = 829

Calculate the square root of the right side: 28.792360098

Break this problem into two subproblems by setting 
(n + -35) equal to 28.792360098 and -28.792360098.

Subproblem 1

n + -35 = 28.792360098 Simplifying n + -35 = 28.792360098 Reorder the terms: -35 + n = 28.792360098 Solving -35 + n = 28.792360098 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '35' to each side of the equation. -35 + 35 + n = 28.792360098 + 35 Combine like terms: -35 + 35 = 0 0 + n = 28.792360098 + 35 n = 28.792360098 + 35 Combine like terms: 28.792360098 + 35 = 63.792360098 n = 63.792360098 Simplifying n = 63.792360098

Subproblem 2

n + -35 = -28.792360098 Simplifying n + -35 = -28.792360098 Reorder the terms: -35 + n = -28.792360098 Solving -35 + n = -28.792360098 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '35' to each side of the equation. -35 + 35 + n = -28.792360098 + 35 Combine like terms: -35 + 35 = 0 0 + n = -28.792360098 + 35 n = -28.792360098 + 35 Combine like terms: -28.792360098 + 35 = 6.207639902 n = 6.207639902 Simplifying n = 6.207639902

Solution

The solution to the problem is based on the solutions from the subproblems. n = {63.792360098, 6.207639902}

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