(n+4)(n-7)=30

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Solution for (n+4)(n-7)=30 equation:


Simplifying
(n + 4)(n + -7) = 30

Reorder the terms:
(4 + n)(n + -7) = 30

Reorder the terms:
(4 + n)(-7 + n) = 30

Multiply (4 + n) * (-7 + n)
(4(-7 + n) + n(-7 + n)) = 30
((-7 * 4 + n * 4) + n(-7 + n)) = 30
((-28 + 4n) + n(-7 + n)) = 30
(-28 + 4n + (-7 * n + n * n)) = 30
(-28 + 4n + (-7n + n2)) = 30

Combine like terms: 4n + -7n = -3n
(-28 + -3n + n2) = 30

Solving
-28 + -3n + n2 = 30

Solving for variable 'n'.

Reorder the terms:
-28 + -30 + -3n + n2 = 30 + -30

Combine like terms: -28 + -30 = -58
-58 + -3n + n2 = 30 + -30

Combine like terms: 30 + -30 = 0
-58 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '58' to each side of the equation.
-58 + -3n + 58 + n2 = 0 + 58

Reorder the terms:
-58 + 58 + -3n + n2 = 0 + 58

Combine like terms: -58 + 58 = 0
0 + -3n + n2 = 0 + 58
-3n + n2 = 0 + 58

Combine like terms: 0 + 58 = 58
-3n + n2 = 58

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 58 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 58 + 2.25

Combine like terms: 58 + 2.25 = 60.25
2.25 + -3n + n2 = 60.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 60.25

Calculate the square root of the right side: 7.762087348

Break this problem into two subproblems by setting 
(n + -1.5) equal to 7.762087348 and -7.762087348.

Subproblem 1

n + -1.5 = 7.762087348 Simplifying n + -1.5 = 7.762087348 Reorder the terms: -1.5 + n = 7.762087348 Solving -1.5 + n = 7.762087348 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 7.762087348 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 7.762087348 + 1.5 n = 7.762087348 + 1.5 Combine like terms: 7.762087348 + 1.5 = 9.262087348 n = 9.262087348 Simplifying n = 9.262087348

Subproblem 2

n + -1.5 = -7.762087348 Simplifying n + -1.5 = -7.762087348 Reorder the terms: -1.5 + n = -7.762087348 Solving -1.5 + n = -7.762087348 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -7.762087348 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -7.762087348 + 1.5 n = -7.762087348 + 1.5 Combine like terms: -7.762087348 + 1.5 = -6.262087348 n = -6.262087348 Simplifying n = -6.262087348

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.262087348, -6.262087348}

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