x^2+x+8=78

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Solution for x^2+x+8=78 equation:


Simplifying
x2 + x + 8 = 78

Reorder the terms:
8 + x + x2 = 78

Solving
8 + x + x2 = 78

Solving for variable 'x'.

Reorder the terms:
8 + -78 + x + x2 = 78 + -78

Combine like terms: 8 + -78 = -70
-70 + x + x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-70 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '70' to each side of the equation.
-70 + x + 70 + x2 = 0 + 70

Reorder the terms:
-70 + 70 + x + x2 = 0 + 70

Combine like terms: -70 + 70 = 0
0 + x + x2 = 0 + 70
x + x2 = 0 + 70

Combine like terms: 0 + 70 = 70
x + x2 = 70

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 = 70 + 0.25

Reorder the terms:
0.25 + x + x2 = 70 + 0.25

Combine like terms: 70 + 0.25 = 70.25
0.25 + x + x2 = 70.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 70.25

Calculate the square root of the right side: 8.381527307

Break this problem into two subproblems by setting 
(x + 0.5) equal to 8.381527307 and -8.381527307.

Subproblem 1

x + 0.5 = 8.381527307 Simplifying x + 0.5 = 8.381527307 Reorder the terms: 0.5 + x = 8.381527307 Solving 0.5 + x = 8.381527307 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 = 8.381527307 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.381527307 + -0.5 x = 8.381527307 + -0.5 Combine like terms: 8.381527307 + -0.5 = 7.881527307 x = 7.881527307 Simplifying x = 7.881527307

Subproblem 2

x + 0.5 = -8.381527307 Simplifying x + 0.5 = -8.381527307 Reorder the terms: 0.5 + x = -8.381527307 Solving 0.5 + x = -8.381527307 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 = -8.381527307 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.381527307 + -0.5 x = -8.381527307 + -0.5 Combine like terms: -8.381527307 + -0.5 = -8.881527307 x = -8.881527307 Simplifying x = -8.881527307

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.881527307, -8.881527307}

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