66-x^2+x=8

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Solution for 66-x^2+x=8 equation:


Simplifying
66 + -1x2 + x = 8

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

Solving
66 + x + -1x2 = 8

Solving for variable 'x'.

Reorder the terms:
66 + -8 + x + -1x2 = 8 + -8

Combine like terms: 66 + -8 = 58
58 + x + -1x2 = 8 + -8

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

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-58 + -1x + x2 = 0

Move the constant term to the right:

Add '58' to each side of the equation.
-58 + -1x + 58 + x2 = 0 + 58

Reorder the terms:
-58 + 58 + -1x + x2 = 0 + 58

Combine like terms: -58 + 58 = 0
0 + -1x + x2 = 0 + 58
-1x + x2 = 0 + 58

Combine like terms: 0 + 58 = 58
-1x + x2 = 58

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.
-1x + 0.25 + x2 = 58 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 58 + 0.25

Combine like terms: 58 + 0.25 = 58.25
0.25 + -1x + x2 = 58.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 58.25

Calculate the square root of the right side: 7.632168761

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.632168761 and -7.632168761.

Subproblem 1

x + 0.5 = 7.632168761 Simplifying x + 0.5 = 7.632168761 Reorder the terms: 0.5 + x = 7.632168761 Solving 0.5 + x = 7.632168761 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 = 7.632168761 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.632168761 + -0.5 x = 7.632168761 + -0.5 Combine like terms: 7.632168761 + -0.5 = 7.132168761 x = 7.132168761 Simplifying x = 7.132168761

Subproblem 2

x + 0.5 = -7.632168761 Simplifying x + 0.5 = -7.632168761 Reorder the terms: 0.5 + x = -7.632168761 Solving 0.5 + x = -7.632168761 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 = -7.632168761 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.632168761 + -0.5 x = -7.632168761 + -0.5 Combine like terms: -7.632168761 + -0.5 = -8.132168761 x = -8.132168761 Simplifying x = -8.132168761

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.132168761, -8.132168761}

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