n^2+2n+5=1542

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Solution for n^2+2n+5=1542 equation:


Simplifying
n2 + 2n + 5 = 1542

Reorder the terms:
5 + 2n + n2 = 1542

Solving
5 + 2n + n2 = 1542

Solving for variable 'n'.

Reorder the terms:
5 + -1542 + 2n + n2 = 1542 + -1542

Combine like terms: 5 + -1542 = -1537
-1537 + 2n + n2 = 1542 + -1542

Combine like terms: 1542 + -1542 = 0
-1537 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1537' to each side of the equation.
-1537 + 2n + 1537 + n2 = 0 + 1537

Reorder the terms:
-1537 + 1537 + 2n + n2 = 0 + 1537

Combine like terms: -1537 + 1537 = 0
0 + 2n + n2 = 0 + 1537
2n + n2 = 0 + 1537

Combine like terms: 0 + 1537 = 1537
2n + n2 = 1537

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 1537 + 1

Reorder the terms:
1 + 2n + n2 = 1537 + 1

Combine like terms: 1537 + 1 = 1538
1 + 2n + n2 = 1538

Factor a perfect square on the left side:
(n + 1)(n + 1) = 1538

Calculate the square root of the right side: 39.217343102

Break this problem into two subproblems by setting 
(n + 1) equal to 39.217343102 and -39.217343102.

Subproblem 1

n + 1 = 39.217343102 Simplifying n + 1 = 39.217343102 Reorder the terms: 1 + n = 39.217343102 Solving 1 + n = 39.217343102 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 39.217343102 + -1 Combine like terms: 1 + -1 = 0 0 + n = 39.217343102 + -1 n = 39.217343102 + -1 Combine like terms: 39.217343102 + -1 = 38.217343102 n = 38.217343102 Simplifying n = 38.217343102

Subproblem 2

n + 1 = -39.217343102 Simplifying n + 1 = -39.217343102 Reorder the terms: 1 + n = -39.217343102 Solving 1 + n = -39.217343102 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -39.217343102 + -1 Combine like terms: 1 + -1 = 0 0 + n = -39.217343102 + -1 n = -39.217343102 + -1 Combine like terms: -39.217343102 + -1 = -40.217343102 n = -40.217343102 Simplifying n = -40.217343102

Solution

The solution to the problem is based on the solutions from the subproblems. n = {38.217343102, -40.217343102}

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