(x-6)-(x^2-5x-24)=1

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Solution for (x-6)-(x^2-5x-24)=1 equation:


Simplifying
(x + -6) + -1(x2 + -5x + -24) = 1

Reorder the terms:
(-6 + x) + -1(x2 + -5x + -24) = 1

Remove parenthesis around (-6 + x)
-6 + x + -1(x2 + -5x + -24) = 1

Reorder the terms:
-6 + x + -1(-24 + -5x + x2) = 1
-6 + x + (-24 * -1 + -5x * -1 + x2 * -1) = 1
-6 + x + (24 + 5x + -1x2) = 1

Reorder the terms:
-6 + 24 + x + 5x + -1x2 = 1

Combine like terms: -6 + 24 = 18
18 + x + 5x + -1x2 = 1

Combine like terms: x + 5x = 6x
18 + 6x + -1x2 = 1

Solving
18 + 6x + -1x2 = 1

Solving for variable 'x'.

Reorder the terms:
18 + -1 + 6x + -1x2 = 1 + -1

Combine like terms: 18 + -1 = 17
17 + 6x + -1x2 = 1 + -1

Combine like terms: 1 + -1 = 0
17 + 6x + -1x2 = 0

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

Divide each side by '-1'.
-17 + -6x + x2 = 0

Move the constant term to the right:

Add '17' to each side of the equation.
-17 + -6x + 17 + x2 = 0 + 17

Reorder the terms:
-17 + 17 + -6x + x2 = 0 + 17

Combine like terms: -17 + 17 = 0
0 + -6x + x2 = 0 + 17
-6x + x2 = 0 + 17

Combine like terms: 0 + 17 = 17
-6x + x2 = 17

The x term is -6x.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6x + 9 + x2 = 17 + 9

Reorder the terms:
9 + -6x + x2 = 17 + 9

Combine like terms: 17 + 9 = 26
9 + -6x + x2 = 26

Factor a perfect square on the left side:
(x + -3)(x + -3) = 26

Calculate the square root of the right side: 5.099019514

Break this problem into two subproblems by setting 
(x + -3) equal to 5.099019514 and -5.099019514.

Subproblem 1

x + -3 = 5.099019514 Simplifying x + -3 = 5.099019514 Reorder the terms: -3 + x = 5.099019514 Solving -3 + x = 5.099019514 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 5.099019514 + 3 Combine like terms: -3 + 3 = 0 0 + x = 5.099019514 + 3 x = 5.099019514 + 3 Combine like terms: 5.099019514 + 3 = 8.099019514 x = 8.099019514 Simplifying x = 8.099019514

Subproblem 2

x + -3 = -5.099019514 Simplifying x + -3 = -5.099019514 Reorder the terms: -3 + x = -5.099019514 Solving -3 + x = -5.099019514 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = -5.099019514 + 3 Combine like terms: -3 + 3 = 0 0 + x = -5.099019514 + 3 x = -5.099019514 + 3 Combine like terms: -5.099019514 + 3 = -2.099019514 x = -2.099019514 Simplifying x = -2.099019514

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.099019514, -2.099019514}

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