(n+2)(n-5)=52

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Solution for (n+2)(n-5)=52 equation:


Simplifying
(n + 2)(n + -5) = 52

Reorder the terms:
(2 + n)(n + -5) = 52

Reorder the terms:
(2 + n)(-5 + n) = 52

Multiply (2 + n) * (-5 + n)
(2(-5 + n) + n(-5 + n)) = 52
((-5 * 2 + n * 2) + n(-5 + n)) = 52
((-10 + 2n) + n(-5 + n)) = 52
(-10 + 2n + (-5 * n + n * n)) = 52
(-10 + 2n + (-5n + n2)) = 52

Combine like terms: 2n + -5n = -3n
(-10 + -3n + n2) = 52

Solving
-10 + -3n + n2 = 52

Solving for variable 'n'.

Reorder the terms:
-10 + -52 + -3n + n2 = 52 + -52

Combine like terms: -10 + -52 = -62
-62 + -3n + n2 = 52 + -52

Combine like terms: 52 + -52 = 0
-62 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '62' to each side of the equation.
-62 + -3n + 62 + n2 = 0 + 62

Reorder the terms:
-62 + 62 + -3n + n2 = 0 + 62

Combine like terms: -62 + 62 = 0
0 + -3n + n2 = 0 + 62
-3n + n2 = 0 + 62

Combine like terms: 0 + 62 = 62
-3n + n2 = 62

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 = 62 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 62 + 2.25

Combine like terms: 62 + 2.25 = 64.25
2.25 + -3n + n2 = 64.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 64.25

Calculate the square root of the right side: 8.015609771

Break this problem into two subproblems by setting 
(n + -1.5) equal to 8.015609771 and -8.015609771.

Subproblem 1

n + -1.5 = 8.015609771 Simplifying n + -1.5 = 8.015609771 Reorder the terms: -1.5 + n = 8.015609771 Solving -1.5 + n = 8.015609771 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 = 8.015609771 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 8.015609771 + 1.5 n = 8.015609771 + 1.5 Combine like terms: 8.015609771 + 1.5 = 9.515609771 n = 9.515609771 Simplifying n = 9.515609771

Subproblem 2

n + -1.5 = -8.015609771 Simplifying n + -1.5 = -8.015609771 Reorder the terms: -1.5 + n = -8.015609771 Solving -1.5 + n = -8.015609771 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 = -8.015609771 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -8.015609771 + 1.5 n = -8.015609771 + 1.5 Combine like terms: -8.015609771 + 1.5 = -6.515609771 n = -6.515609771 Simplifying n = -6.515609771

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.515609771, -6.515609771}

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