8x=-x^2-1

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Solution for 8x=-x^2-1 equation:


Simplifying
8x = -1x2 + -1

Reorder the terms:
8x = -1 + -1x2

Solving
8x = -1 + -1x2

Solving for variable 'x'.

Reorder the terms:
1 + 8x + x2 = -1 + -1x2 + 1 + x2

Reorder the terms:
1 + 8x + x2 = -1 + 1 + -1x2 + x2

Combine like terms: -1 + 1 = 0
1 + 8x + x2 = 0 + -1x2 + x2
1 + 8x + x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
1 + 8x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 8x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 8x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 8x + x2 = 0 + -1
8x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
8x + x2 = -1

The x term is 8x.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8x + 16 + x2 = -1 + 16

Reorder the terms:
16 + 8x + x2 = -1 + 16

Combine like terms: -1 + 16 = 15
16 + 8x + x2 = 15

Factor a perfect square on the left side:
(x + 4)(x + 4) = 15

Calculate the square root of the right side: 3.872983346

Break this problem into two subproblems by setting 
(x + 4) equal to 3.872983346 and -3.872983346.

Subproblem 1

x + 4 = 3.872983346 Simplifying x + 4 = 3.872983346 Reorder the terms: 4 + x = 3.872983346 Solving 4 + x = 3.872983346 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = 3.872983346 + -4 Combine like terms: 4 + -4 = 0 0 + x = 3.872983346 + -4 x = 3.872983346 + -4 Combine like terms: 3.872983346 + -4 = -0.127016654 x = -0.127016654 Simplifying x = -0.127016654

Subproblem 2

x + 4 = -3.872983346 Simplifying x + 4 = -3.872983346 Reorder the terms: 4 + x = -3.872983346 Solving 4 + x = -3.872983346 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = -3.872983346 + -4 Combine like terms: 4 + -4 = 0 0 + x = -3.872983346 + -4 x = -3.872983346 + -4 Combine like terms: -3.872983346 + -4 = -7.872983346 x = -7.872983346 Simplifying x = -7.872983346

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.127016654, -7.872983346}

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