x^2+x+1=45

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Solution for x^2+x+1=45 equation:


Simplifying
x2 + x + 1 = 45

Reorder the terms:
1 + x + x2 = 45

Solving
1 + x + x2 = 45

Solving for variable 'x'.

Reorder the terms:
1 + -45 + x + x2 = 45 + -45

Combine like terms: 1 + -45 = -44
-44 + x + x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-44 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '44' to each side of the equation.
-44 + x + 44 + x2 = 0 + 44

Reorder the terms:
-44 + 44 + x + x2 = 0 + 44

Combine like terms: -44 + 44 = 0
0 + x + x2 = 0 + 44
x + x2 = 0 + 44

Combine like terms: 0 + 44 = 44
x + x2 = 44

The x term is x.  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.
x + 0.25 + x2 = 44 + 0.25

Reorder the terms:
0.25 + x + x2 = 44 + 0.25

Combine like terms: 44 + 0.25 = 44.25
0.25 + x + x2 = 44.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 44.25

Calculate the square root of the right side: 6.652067348

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.652067348 and -6.652067348.

Subproblem 1

x + 0.5 = 6.652067348 Simplifying x + 0.5 = 6.652067348 Reorder the terms: 0.5 + x = 6.652067348 Solving 0.5 + x = 6.652067348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 6.652067348 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.652067348 + -0.5 x = 6.652067348 + -0.5 Combine like terms: 6.652067348 + -0.5 = 6.152067348 x = 6.152067348 Simplifying x = 6.152067348

Subproblem 2

x + 0.5 = -6.652067348 Simplifying x + 0.5 = -6.652067348 Reorder the terms: 0.5 + x = -6.652067348 Solving 0.5 + x = -6.652067348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -6.652067348 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.652067348 + -0.5 x = -6.652067348 + -0.5 Combine like terms: -6.652067348 + -0.5 = -7.152067348 x = -7.152067348 Simplifying x = -7.152067348

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.152067348, -7.152067348}

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