x+x^2+(x^2-6)=69

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Solution for x+x^2+(x^2-6)=69 equation:


Simplifying
x + x2 + (x2 + -6) = 69

Reorder the terms:
x + x2 + (-6 + x2) = 69

Remove parenthesis around (-6 + x2)
x + x2 + -6 + x2 = 69

Reorder the terms:
-6 + x + x2 + x2 = 69

Combine like terms: x2 + x2 = 2x2
-6 + x + 2x2 = 69

Solving
-6 + x + 2x2 = 69

Solving for variable 'x'.

Reorder the terms:
-6 + -69 + x + 2x2 = 69 + -69

Combine like terms: -6 + -69 = -75
-75 + x + 2x2 = 69 + -69

Combine like terms: 69 + -69 = 0
-75 + x + 2x2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-37.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '37.5' to each side of the equation.
-37.5 + 0.5x + 37.5 + x2 = 0 + 37.5

Reorder the terms:
-37.5 + 37.5 + 0.5x + x2 = 0 + 37.5

Combine like terms: -37.5 + 37.5 = 0.0
0.0 + 0.5x + x2 = 0 + 37.5
0.5x + x2 = 0 + 37.5

Combine like terms: 0 + 37.5 = 37.5
0.5x + x2 = 37.5

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.
0.5x + 0.25 + x2 = 37.5 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 37.5 + 0.25

Combine like terms: 37.5 + 0.25 = 37.75
0.25 + 0.5x + x2 = 37.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 37.75

Calculate the square root of the right side: 6.144102864

Break this problem into two subproblems by setting 
(x + 0.5) equal to 6.144102864 and -6.144102864.

Subproblem 1

x + 0.5 = 6.144102864 Simplifying x + 0.5 = 6.144102864 Reorder the terms: 0.5 + x = 6.144102864 Solving 0.5 + x = 6.144102864 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 = 6.144102864 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.144102864 + -0.5 x = 6.144102864 + -0.5 Combine like terms: 6.144102864 + -0.5 = 5.644102864 x = 5.644102864 Simplifying x = 5.644102864

Subproblem 2

x + 0.5 = -6.144102864 Simplifying x + 0.5 = -6.144102864 Reorder the terms: 0.5 + x = -6.144102864 Solving 0.5 + x = -6.144102864 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 = -6.144102864 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.144102864 + -0.5 x = -6.144102864 + -0.5 Combine like terms: -6.144102864 + -0.5 = -6.644102864 x = -6.644102864 Simplifying x = -6.644102864

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.644102864, -6.644102864}

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