-x^2+23x-24=0

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Solution for -x^2+23x-24=0 equation:


Simplifying
-1x2 + 23x + -24 = 0

Reorder the terms:
-24 + 23x + -1x2 = 0

Solving
-24 + 23x + -1x2 = 0

Solving for variable 'x'.

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

Divide each side by '-1'.
24 + -23x + x2 = 0

Move the constant term to the right:

Add '-24' to each side of the equation.
24 + -23x + -24 + x2 = 0 + -24

Reorder the terms:
24 + -24 + -23x + x2 = 0 + -24

Combine like terms: 24 + -24 = 0
0 + -23x + x2 = 0 + -24
-23x + x2 = 0 + -24

Combine like terms: 0 + -24 = -24
-23x + x2 = -24

The x term is -23x.  Take half its coefficient (-11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
-23x + 132.25 + x2 = -24 + 132.25

Reorder the terms:
132.25 + -23x + x2 = -24 + 132.25

Combine like terms: -24 + 132.25 = 108.25
132.25 + -23x + x2 = 108.25

Factor a perfect square on the left side:
(x + -11.5)(x + -11.5) = 108.25

Calculate the square root of the right side: 10.404326023

Break this problem into two subproblems by setting 
(x + -11.5) equal to 10.404326023 and -10.404326023.

Subproblem 1

x + -11.5 = 10.404326023 Simplifying x + -11.5 = 10.404326023 Reorder the terms: -11.5 + x = 10.404326023 Solving -11.5 + x = 10.404326023 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + x = 10.404326023 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + x = 10.404326023 + 11.5 x = 10.404326023 + 11.5 Combine like terms: 10.404326023 + 11.5 = 21.904326023 x = 21.904326023 Simplifying x = 21.904326023

Subproblem 2

x + -11.5 = -10.404326023 Simplifying x + -11.5 = -10.404326023 Reorder the terms: -11.5 + x = -10.404326023 Solving -11.5 + x = -10.404326023 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + x = -10.404326023 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + x = -10.404326023 + 11.5 x = -10.404326023 + 11.5 Combine like terms: -10.404326023 + 11.5 = 1.095673977 x = 1.095673977 Simplifying x = 1.095673977

Solution

The solution to the problem is based on the solutions from the subproblems. x = {21.904326023, 1.095673977}

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