(n+4)(n-3)=22

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Solution for (n+4)(n-3)=22 equation:


Simplifying
(n + 4)(n + -3) = 22

Reorder the terms:
(4 + n)(n + -3) = 22

Reorder the terms:
(4 + n)(-3 + n) = 22

Multiply (4 + n) * (-3 + n)
(4(-3 + n) + n(-3 + n)) = 22
((-3 * 4 + n * 4) + n(-3 + n)) = 22
((-12 + 4n) + n(-3 + n)) = 22
(-12 + 4n + (-3 * n + n * n)) = 22
(-12 + 4n + (-3n + n2)) = 22

Combine like terms: 4n + -3n = 1n
(-12 + 1n + n2) = 22

Solving
-12 + 1n + n2 = 22

Solving for variable 'n'.

Reorder the terms:
-12 + -22 + 1n + n2 = 22 + -22

Combine like terms: -12 + -22 = -34
-34 + 1n + n2 = 22 + -22

Combine like terms: 22 + -22 = 0
-34 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '34' to each side of the equation.
-34 + 1n + 34 + n2 = 0 + 34

Reorder the terms:
-34 + 34 + 1n + n2 = 0 + 34

Combine like terms: -34 + 34 = 0
0 + 1n + n2 = 0 + 34
1n + n2 = 0 + 34

Combine like terms: 0 + 34 = 34
1n + n2 = 34

The n term is 1n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 34 + 0.25

Reorder the terms:
0.25 + n + n2 = 34 + 0.25

Combine like terms: 34 + 0.25 = 34.25
0.25 + n + n2 = 34.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 34.25

Calculate the square root of the right side: 5.852349955

Break this problem into two subproblems by setting 
(n + 0.5) equal to 5.852349955 and -5.852349955.

Subproblem 1

n + 0.5 = 5.852349955 Simplifying n + 0.5 = 5.852349955 Reorder the terms: 0.5 + n = 5.852349955 Solving 0.5 + n = 5.852349955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 5.852349955 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 5.852349955 + -0.5 n = 5.852349955 + -0.5 Combine like terms: 5.852349955 + -0.5 = 5.352349955 n = 5.352349955 Simplifying n = 5.352349955

Subproblem 2

n + 0.5 = -5.852349955 Simplifying n + 0.5 = -5.852349955 Reorder the terms: 0.5 + n = -5.852349955 Solving 0.5 + n = -5.852349955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -5.852349955 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -5.852349955 + -0.5 n = -5.852349955 + -0.5 Combine like terms: -5.852349955 + -0.5 = -6.352349955 n = -6.352349955 Simplifying n = -6.352349955

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.352349955, -6.352349955}

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