x*x+x=45

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Solution for x*x+x=45 equation:


Simplifying
x * x + x = 45

Multiply x * x
x2 + x = 45

Reorder the terms:
x + x2 = 45

Solving
x + x2 = 45

Solving for variable 'x'.

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

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

Begin completing the square.

Move the constant term to the right:

Add '45' to each side of the equation.
-45 + x + 45 + x2 = 0 + 45

Reorder the terms:
-45 + 45 + x + x2 = 0 + 45

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

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

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 = 45 + 0.25

Reorder the terms:
0.25 + x + x2 = 45 + 0.25

Combine like terms: 45 + 0.25 = 45.25
0.25 + x + x2 = 45.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 45.25

Calculate the square root of the right side: 6.726812024

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.726812024 and -6.726812024.

Subproblem 1

x + 0.5 = 6.726812024 Simplifying x + 0.5 = 6.726812024 Reorder the terms: 0.5 + x = 6.726812024 Solving 0.5 + x = 6.726812024 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 = 6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.726812024 + -0.5 x = 6.726812024 + -0.5 Combine like terms: 6.726812024 + -0.5 = 6.226812024 x = 6.226812024 Simplifying x = 6.226812024

Subproblem 2

x + 0.5 = -6.726812024 Simplifying x + 0.5 = -6.726812024 Reorder the terms: 0.5 + x = -6.726812024 Solving 0.5 + x = -6.726812024 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 = -6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.726812024 + -0.5 x = -6.726812024 + -0.5 Combine like terms: -6.726812024 + -0.5 = -7.226812024 x = -7.226812024 Simplifying x = -7.226812024

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.226812024, -7.226812024}

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