n^2+13n-22=7

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Solution for n^2+13n-22=7 equation:


Simplifying
n2 + 13n + -22 = 7

Reorder the terms:
-22 + 13n + n2 = 7

Solving
-22 + 13n + n2 = 7

Solving for variable 'n'.

Reorder the terms:
-22 + -7 + 13n + n2 = 7 + -7

Combine like terms: -22 + -7 = -29
-29 + 13n + n2 = 7 + -7

Combine like terms: 7 + -7 = 0
-29 + 13n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '29' to each side of the equation.
-29 + 13n + 29 + n2 = 0 + 29

Reorder the terms:
-29 + 29 + 13n + n2 = 0 + 29

Combine like terms: -29 + 29 = 0
0 + 13n + n2 = 0 + 29
13n + n2 = 0 + 29

Combine like terms: 0 + 29 = 29
13n + n2 = 29

The n term is 13n.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13n + 42.25 + n2 = 29 + 42.25

Reorder the terms:
42.25 + 13n + n2 = 29 + 42.25

Combine like terms: 29 + 42.25 = 71.25
42.25 + 13n + n2 = 71.25

Factor a perfect square on the left side:
(n + 6.5)(n + 6.5) = 71.25

Calculate the square root of the right side: 8.440971508

Break this problem into two subproblems by setting 
(n + 6.5) equal to 8.440971508 and -8.440971508.

Subproblem 1

n + 6.5 = 8.440971508 Simplifying n + 6.5 = 8.440971508 Reorder the terms: 6.5 + n = 8.440971508 Solving 6.5 + n = 8.440971508 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = 8.440971508 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = 8.440971508 + -6.5 n = 8.440971508 + -6.5 Combine like terms: 8.440971508 + -6.5 = 1.940971508 n = 1.940971508 Simplifying n = 1.940971508

Subproblem 2

n + 6.5 = -8.440971508 Simplifying n + 6.5 = -8.440971508 Reorder the terms: 6.5 + n = -8.440971508 Solving 6.5 + n = -8.440971508 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = -8.440971508 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = -8.440971508 + -6.5 n = -8.440971508 + -6.5 Combine like terms: -8.440971508 + -6.5 = -14.940971508 n = -14.940971508 Simplifying n = -14.940971508

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.940971508, -14.940971508}

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